The algebra of the box–spline
Claudio Procesi.
Hanoi, January 2011
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
BASIC INPUT
The basic input of this theory is a
(real, sometimes integer) n × m matrix A.
We always think of A as a LIST of vectors in V = Rn , its columns:
A := (a1 , . . . , am )
Constrain
We assume that 0 is NOT in the convex hull of its columns.
From A we make several constructions, algebraic, combinatorial,
analytic etc..
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
For a complete treatment see
Series: Universitext
1st Edition., 2010, XXII, 381
p. 19 illus., 4 in color., Softcover
ISBN: 978-0-387-78962-0
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
CONVEX POLYTOPES
We begin with a system of linear equations:
m
X
ai xi = b,
or
Ax = b,
A := (a1 , . . . , am )
(1)
i=1
The
columns ai , b are vectors with n coordinates
(aj,i , bj , j = 1, . . . , n).
We assume that the matrix A is real and 0 is NOT in the convex
hull of its columns.
We are interested in dealing with two
fundamental problems.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
CONVEX POLYTOPES
We begin with a system of linear equations:
m
X
ai xi = b,
or
Ax = b,
A := (a1 , . . . , am )
(1)
i=1
The
columns ai , b are vectors with n coordinates
(aj,i , bj , j = 1, . . . , n).
We assume that the matrix A is real and 0 is NOT in the convex
hull of its columns.
We are interested in dealing with two
fundamental problems.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
CONVEX POLYTOPES
We begin with a system of linear equations:
m
X
ai xi = b,
or
Ax = b,
A := (a1 , . . . , am )
(1)
i=1
The
columns ai , b are vectors with n coordinates
(aj,i , bj , j = 1, . . . , n).
We assume that the matrix A is real and 0 is NOT in the convex
hull of its columns.
We are interested in dealing with two
fundamental problems.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
CONVEX POLYTOPES
We begin with a system of linear equations:
m
X
ai xi = b,
or
Ax = b,
A := (a1 , . . . , am )
(1)
i=1
The
columns ai , b are vectors with n coordinates
(aj,i , bj , j = 1, . . . , n).
We assume that the matrix A is real and 0 is NOT in the convex
hull of its columns.
We are interested in dealing with two
fundamental problems.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Variable polytopes
As in Linear Programming Theory we want to study the
VARIABLE POLYTOPES:
ΠA (b) := {x | Ax = b, xi ≥ 0, ∀i}
which are convex and bounded for every b.
Identify the spaces Ax = b and Ax = 0 then think of ΠA (b) as a
variable polytope in the space
Claudio Procesi.
Ax = 0
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Variable polytopes
As in Linear Programming Theory we want to study the
VARIABLE POLYTOPES:
ΠA (b) := {x | Ax = b, xi ≥ 0, ∀i}
which are convex and bounded for every b.
Identify the spaces Ax = b and Ax = 0 then think of ΠA (b) as a
variable polytope in the space
Claudio Procesi.
Ax = 0
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Variable polytopes
As in Linear Programming Theory we want to study the
VARIABLE POLYTOPES:
ΠA (b) := {x | Ax = b, xi ≥ 0, ∀i}
which are convex and bounded for every b.
Identify the spaces Ax = b and Ax = 0 then think of ΠA (b) as a
variable polytope in the space
Claudio Procesi.
Ax = 0
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The object of study
Basic functions
Set VA (b) to be the volume of ΠA (b).
If A, b have integer coordinates, consider the finite set IA (b)
of points in ΠA (b) with integer coordinates.
Arithmetic case
In this case we want compute the number PA (b) of solutions of
the system in which the coordinates xi are non negative integers.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The object of study
Basic functions
Set VA (b) to be the volume of ΠA (b).
If A, b have integer coordinates, consider the finite set IA (b)
of points in ΠA (b) with integer coordinates.
Arithmetic case
In this case we want compute the number PA (b) of solutions of
the system in which the coordinates xi are non negative integers.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
In other words the vectors a1 , . . . , am define a map:
F : Rm → Rn .
F (t1 , . . . tm ) := t1 a1 + · · · + tm am .
Therefore, in this language, the polytope:
ΠA (b) = F −1 (b) ∩ Rm+ ,
where Rm+ denotes the positive quadrant.
Our goals are therefore:
1
To compute the volume of ΠA (b).
2
If A, b have integer elements, to compute the number PA (b)
of points with integer coordinates in ΠA (b).
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
In other words the vectors a1 , . . . , am define a map:
F : Rm → Rn .
F (t1 , . . . tm ) := t1 a1 + · · · + tm am .
Therefore, in this language, the polytope:
ΠA (b) = F −1 (b) ∩ Rm+ ,
where Rm+ denotes the positive quadrant.
Our goals are therefore:
1
To compute the volume of ΠA (b).
2
If A, b have integer elements, to compute the number PA (b)
of points with integer coordinates in ΠA (b).
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
In other words the vectors a1 , . . . , am define a map:
F : Rm → Rn .
F (t1 , . . . tm ) := t1 a1 + · · · + tm am .
Therefore, in this language, the polytope:
ΠA (b) = F −1 (b) ∩ Rm+ ,
where Rm+ denotes the positive quadrant.
Our goals are therefore:
1
To compute the volume of ΠA (b).
2
If A, b have integer elements, to compute the number PA (b)
of points with integer coordinates in ΠA (b).
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
When A, b have integer elements it is natural to think of an
expression like:
b = t1 a1 + · · · + tm am with ti not negative integers as a:
partition of b with the vectors ai ,
in t1 + t2 + · · · + tm parts, hence the name partition function for
the number PA (b), thought of as a function of the vector b.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
When A, b have integer elements it is natural to think of an
expression like:
b = t1 a1 + · · · + tm am with ti not negative integers as a:
partition of b with the vectors ai ,
in t1 + t2 + · · · + tm parts, hence the name partition function for
the number PA (b), thought of as a function of the vector b.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Some combinatorial geometry
The pictures are taken from the book:
C. De Boor, K. Höllig, S. Riemenschneider,
Box splines
Applied Mathematical Sciences 98 (1993).
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The basic cone
Obviously, the set of vectors b such that ΠA (b) is not empty is
equal, by definition, to the set:
The cone generated by A
CA := {
m
X
xi ai | xi ≥ 0}
i=1
CA is a convex cone in Rn .
By assumption its non zero elements lie entirely in the interior of a
half space.
CA is usually said to be a pointed cone and 0 is its vertex.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The basic cone
Obviously, the set of vectors b such that ΠA (b) is not empty is
equal, by definition, to the set:
The cone generated by A
CA := {
m
X
xi ai | xi ≥ 0}
i=1
CA is a convex cone in Rn .
By assumption its non zero elements lie entirely in the interior of a
half space.
CA is usually said to be a pointed cone and 0 is its vertex.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The basic cone
Obviously, the set of vectors b such that ΠA (b) is not empty is
equal, by definition, to the set:
The cone generated by A
CA := {
m
X
xi ai | xi ≥ 0}
i=1
CA is a convex cone in Rn .
By assumption its non zero elements lie entirely in the interior of a
half space.
CA is usually said to be a pointed cone and 0 is its vertex.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A 2–DIMENSIONAL EXAMPLE
1 1 0
A=
1 0 1
−1
1
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O
?
??

?? 
? 
/
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A 2–DIMENSIONAL EXAMPLE
the associated cone C (A) has three big cells
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Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
It is useful to consider also the dual cone ĈA of CA .
In the dual space it consists of the (row) vectors that have non
negative scalar product with all vectors ai . I.e. with all vectors of
the cone CA .
ĈA is a cone with non empty interior.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
DUAL CONES
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Claudio Procesi.
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Topics in Hyperplane Arrangements, Polytopes and Box-Splines
DUAL CONES
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Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
DUAL CONES
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Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Some preliminaries
In order to attack the problem of computing volumes and partition
functions we need to
describe some combinatorial geometry of the cone C (A)
generated by the columns of A
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Some preliminaries
In order to attack the problem of computing volumes and partition
functions we need to
describe some combinatorial geometry of the cone C (A)
generated by the columns of A
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Some preliminaries
In order to attack the problem of computing volumes and partition
functions we need to
describe some combinatorial geometry of the cone C (A)
generated by the columns of A
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Some preliminaries
In order to attack the problem of computing volumes and partition
functions we need to
describe some combinatorial geometry of the cone C (A)
generated by the columns of A
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Some preliminaries
In order to attack the problem of computing volumes and partition
functions we need to
describe some combinatorial geometry of the cone C (A)
generated by the columns of A
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Some preliminaries
In order to attack the problem of computing volumes and partition
functions we need to
describe some combinatorial geometry of the cone C (A)
generated by the columns of A
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Some preliminaries
In order to attack the problem of computing volumes and partition
functions we need to
describe some combinatorial geometry of the cone C (A)
generated by the columns of A
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Some preliminaries
In order to attack the problem of computing volumes and partition
functions we need to
describe some combinatorial geometry of the cone C (A)
generated by the columns of A
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Some preliminaries
In order to attack the problem of computing volumes and partition
functions we need to
describe some combinatorial geometry of the cone C (A)
generated by the columns of A
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Some preliminaries
In order to attack the problem of computing volumes and partition
functions we need to
describe some combinatorial geometry of the cone C (A)
generated by the columns of A
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Some intuitive pictures
We want to decompose the cone C (A) into
its singular and regular points.
big cells and define
We do everything on a transversal section, where the cone looks
like a bounded convex polytope and then project.
Example
Let us start with some pictures where A is the list of positive roots
for type A3 .
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Some intuitive pictures
We want to decompose the cone C (A) into
its singular and regular points.
big cells and define
We do everything on a transversal section, where the cone looks
like a bounded convex polytope and then project.
Example
Let us start with some pictures where A is the list of positive roots
for type A3 .
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE Type A3 in section (big cells):
α2
DD
DD
zz
DD
zz
z
DD
z
z
DD
z
z
DD
z
z
DD
z
z
DD
zz
D
z
z
α1 + α2P
α2 + 6α3
PPP
66
nnn
PPP
66
nnn
PPP
n
n
66
PP
nnn
n
66
α1 + α2 + αV3
66
V
h
VVVV
66
hhh
h
V
h
V
h
VVVV
6
hhh
h
V
h
V
h
VVVV 66
h
VVV
hhhhhhh
α3
α1
We have 7 big cells.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE Type A3 in section (small cells):
α2 D
DD
zz
DD
z
DD
zz
z
DD
z
z
DD
z
z
DD
z
z
DD
z
z
DD
z
zz
α2 + 6α3
α1 + α2P
PPP
66
nnn
P
n
P
n
66
PPP
nn
n
P
n
66
PP
nnn
66
α1 + α2 + αV3
66
V
h
VVVV
66
hhh
h
V
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h
VVVV
6
hhh
h
V
h
V
h
VVVV 66
h
VVV
hhhhhhh
α1
We have 8 small cells.
Claudio Procesi.
α3
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE 2 in section (big cells):
α2
666
66
66
66
66
66
66
66
66
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α1
We have 3 big cells.
α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE 2 in section (small cells):
α2
6
666
66
66
66
66
66
66
QQQ
mm6m
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m
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mmmm
α1
We have 6 small cells.
Claudio Procesi.
α3
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Geometry of polyhedra
We start with a general remark.
Let be given m points Pi in n dimensional space and X be
their convex envelop. We assume that the points are not
contained in any proper linear subspace.
If we choose in any possible way k ≤ n among these points
that are independent they generate a simplex of dimension
k − 1, let Y be the union of all these simplices.
Y is a closed subset of X of dimension n − 1.
The connected components of X − Y are called
associated to the points.
big cells
The small cells, which we will not use, are obtained by
removing to X all the points laying on the subspaces
generated by such sets of points.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Geometry of polyhedra
We start with a general remark.
Let be given m points Pi in n dimensional space and X be
their convex envelop. We assume that the points are not
contained in any proper linear subspace.
If we choose in any possible way k ≤ n among these points
that are independent they generate a simplex of dimension
k − 1, let Y be the union of all these simplices.
Y is a closed subset of X of dimension n − 1.
The connected components of X − Y are called
associated to the points.
big cells
The small cells, which we will not use, are obtained by
removing to X all the points laying on the subspaces
generated by such sets of points.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Geometry of polyhedra
We start with a general remark.
Let be given m points Pi in n dimensional space and X be
their convex envelop. We assume that the points are not
contained in any proper linear subspace.
If we choose in any possible way k ≤ n among these points
that are independent they generate a simplex of dimension
k − 1, let Y be the union of all these simplices.
Y is a closed subset of X of dimension n − 1.
The connected components of X − Y are called
associated to the points.
big cells
The small cells, which we will not use, are obtained by
removing to X all the points laying on the subspaces
generated by such sets of points.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Geometry of polyhedra
We start with a general remark.
Let be given m points Pi in n dimensional space and X be
their convex envelop. We assume that the points are not
contained in any proper linear subspace.
If we choose in any possible way k ≤ n among these points
that are independent they generate a simplex of dimension
k − 1, let Y be the union of all these simplices.
Y is a closed subset of X of dimension n − 1.
The connected components of X − Y are called
associated to the points.
big cells
The small cells, which we will not use, are obtained by
removing to X all the points laying on the subspaces
generated by such sets of points.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Geometry of polyhedra
We start with a general remark.
Let be given m points Pi in n dimensional space and X be
their convex envelop. We assume that the points are not
contained in any proper linear subspace.
If we choose in any possible way k ≤ n among these points
that are independent they generate a simplex of dimension
k − 1, let Y be the union of all these simplices.
Y is a closed subset of X of dimension n − 1.
The connected components of X − Y are called
associated to the points.
big cells
The small cells, which we will not use, are obtained by
removing to X all the points laying on the subspaces
generated by such sets of points.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Geometry of polyhedra
We start with a general remark.
Let be given m points Pi in n dimensional space and X be
their convex envelop. We assume that the points are not
contained in any proper linear subspace.
If we choose in any possible way k ≤ n among these points
that are independent they generate a simplex of dimension
k − 1, let Y be the union of all these simplices.
Y is a closed subset of X of dimension n − 1.
The connected components of X − Y are called
associated to the points.
big cells
The small cells, which we will not use, are obtained by
removing to X all the points laying on the subspaces
generated by such sets of points.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
PROJECTING A POLYHEDRON TO FORM A CONE
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Same geometry for the cone C (A).
Let be given a list A of vectors ai in n dimensional space, that
generate a pointed cone C (A).
Let us take an hyperplane that intersects the half lines R+ ai
in the points Pi (there exists one by hypothesis).
The cone C (A) is the projection from the origin of the convex
envelop X of the points Pi
Every simplex of dimension ≤ n − 2 projects to a simplicial
cone of dimension ≤ n − 1.
The union of such cones projects Y and the big cells of C (A)
are the projections of the big cells of X minus the origin.
Regular points
The points of the big cells are also called regular, the others
singular. X , Y represent C (A) and its singular points
section.
Claudio Procesi.
in
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Same geometry for the cone C (A).
Let be given a list A of vectors ai in n dimensional space, that
generate a pointed cone C (A).
Let us take an hyperplane that intersects the half lines R+ ai
in the points Pi (there exists one by hypothesis).
The cone C (A) is the projection from the origin of the convex
envelop X of the points Pi
Every simplex of dimension ≤ n − 2 projects to a simplicial
cone of dimension ≤ n − 1.
The union of such cones projects Y and the big cells of C (A)
are the projections of the big cells of X minus the origin.
Regular points
The points of the big cells are also called regular, the others
singular. X , Y represent C (A) and its singular points
section.
Claudio Procesi.
in
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Same geometry for the cone C (A).
Let be given a list A of vectors ai in n dimensional space, that
generate a pointed cone C (A).
Let us take an hyperplane that intersects the half lines R+ ai
in the points Pi (there exists one by hypothesis).
The cone C (A) is the projection from the origin of the convex
envelop X of the points Pi
Every simplex of dimension ≤ n − 2 projects to a simplicial
cone of dimension ≤ n − 1.
The union of such cones projects Y and the big cells of C (A)
are the projections of the big cells of X minus the origin.
Regular points
The points of the big cells are also called regular, the others
singular. X , Y represent C (A) and its singular points
section.
Claudio Procesi.
in
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Same geometry for the cone C (A).
Let be given a list A of vectors ai in n dimensional space, that
generate a pointed cone C (A).
Let us take an hyperplane that intersects the half lines R+ ai
in the points Pi (there exists one by hypothesis).
The cone C (A) is the projection from the origin of the convex
envelop X of the points Pi
Every simplex of dimension ≤ n − 2 projects to a simplicial
cone of dimension ≤ n − 1.
The union of such cones projects Y and the big cells of C (A)
are the projections of the big cells of X minus the origin.
Regular points
The points of the big cells are also called regular, the others
singular. X , Y represent C (A) and its singular points
section.
Claudio Procesi.
in
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Same geometry for the cone C (A).
Let be given a list A of vectors ai in n dimensional space, that
generate a pointed cone C (A).
Let us take an hyperplane that intersects the half lines R+ ai
in the points Pi (there exists one by hypothesis).
The cone C (A) is the projection from the origin of the convex
envelop X of the points Pi
Every simplex of dimension ≤ n − 2 projects to a simplicial
cone of dimension ≤ n − 1.
The union of such cones projects Y and the big cells of C (A)
are the projections of the big cells of X minus the origin.
Regular points
The points of the big cells are also called regular, the others
singular. X , Y represent C (A) and its singular points
section.
Claudio Procesi.
in
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SPLINES TA (x ) =
√
det AAt
−1
VA (x )
The volume
√ of the variable polytope VA (x ) equals, up to the
constant det AAt to the
multivariate spline
that is the function TA (x ) characterized by the formula:
Z
Rn
Z
f (x )TA (x )dx =
Rm
+
f(
m
X
ti ai )dt,
i=1
where f (x ) is any continuous function with compact support.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SPLINES TA (x ) =
√
det AAt
−1
VA (x )
The volume
√ of the variable polytope VA (x ) equals, up to the
constant det AAt to the
multivariate spline
that is the function TA (x ) characterized by the formula:
Z
Rn
Z
f (x )TA (x )dx =
Rm
+
f(
m
X
ti ai )dt,
i=1
where f (x ) is any continuous function with compact support.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SPLINES TA (x ) =
√
det AAt
−1
VA (x )
The volume
√ of the variable polytope VA (x ) equals, up to the
constant det AAt to the
multivariate spline
that is the function TA (x ) characterized by the formula:
Z
Rn
Z
f (x )TA (x )dx =
Rm
+
f(
m
X
ti ai )dt,
i=1
where f (x ) is any continuous function with compact support.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
WARNING
We have given a weak definition for TA (x )
In general
TA (x ) is a tempered distribution, supported on the cone C (A)!
Only when A has maximal rank TA (x ) is a function.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
WARNING
We have given a weak definition for TA (x )
In general
TA (x ) is a tempered distribution, supported on the cone C (A)!
Only when A has maximal rank TA (x ) is a function.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
WARNING
We have given a weak definition for TA (x )
In general
TA (x ) is a tempered distribution, supported on the cone C (A)!
Only when A has maximal rank TA (x ) is a function.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Assume A spans Rn .
Let h be the minimum number of columns that one can remove
from A so that the remaining columns do not span Rn .
The basic function TA (x ) is a spline
1
TA (x ) has support on the cone C (A)
2
TA (x ) is continuous if h ≥ 2
3
In general TA (x ) is of class h − 2
4
TA (x ) coincides with a homogeneous polynomial of degree
m − n on each big cell.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Assume A spans Rn .
Let h be the minimum number of columns that one can remove
from A so that the remaining columns do not span Rn .
The basic function TA (x ) is a spline
1
TA (x ) has support on the cone C (A)
2
TA (x ) is continuous if h ≥ 2
3
In general TA (x ) is of class h − 2
4
TA (x ) coincides with a homogeneous polynomial of degree
m − n on each big cell.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Assume A spans Rn .
Let h be the minimum number of columns that one can remove
from A so that the remaining columns do not span Rn .
The basic function TA (x ) is a spline
1
TA (x ) has support on the cone C (A)
2
TA (x ) is continuous if h ≥ 2
3
In general TA (x ) is of class h − 2
4
TA (x ) coincides with a homogeneous polynomial of degree
m − n on each big cell.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Assume A spans Rn .
Let h be the minimum number of columns that one can remove
from A so that the remaining columns do not span Rn .
The basic function TA (x ) is a spline
1
TA (x ) has support on the cone C (A)
2
TA (x ) is continuous if h ≥ 2
3
In general TA (x ) is of class h − 2
4
TA (x ) coincides with a homogeneous polynomial of degree
m − n on each big cell.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Assume A spans Rn .
Let h be the minimum number of columns that one can remove
from A so that the remaining columns do not span Rn .
The basic function TA (x ) is a spline
1
TA (x ) has support on the cone C (A)
2
TA (x ) is continuous if h ≥ 2
3
In general TA (x ) is of class h − 2
4
TA (x ) coincides with a homogeneous polynomial of degree
m − n on each big cell.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Assume A spans Rn .
Let h be the minimum number of columns that one can remove
from A so that the remaining columns do not span Rn .
The basic function TA (x ) is a spline
1
TA (x ) has support on the cone C (A)
2
TA (x ) is continuous if h ≥ 2
3
In general TA (x ) is of class h − 2
4
TA (x ) coincides with a homogeneous polynomial of degree
m − n on each big cell.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Assume A spans Rn .
Let h be the minimum number of columns that one can remove
from A so that the remaining columns do not span Rn .
The basic function TA (x ) is a spline
1
TA (x ) has support on the cone C (A)
2
TA (x ) is continuous if h ≥ 2
3
In general TA (x ) is of class h − 2
4
TA (x ) coincides with a homogeneous polynomial of degree
m − n on each big cell.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SIMPLE EXAMPLE
1 0 1 A=
0 1 1 ~?
~~
~
~~
/
.~
O
C (A) is the
first quadrant
The three vectors
The function TA (x , y ) is
of class C 0
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
the associated cone C (A) has two big cells
TA is 0 outside the first
















O
?





/
quadrant and on the two cells:
0
x
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
TA is 0 outside the first quadrant and on the two cells:
0
x
O
?




/
y
0
0
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SUMMARIZING
In order to compute TA (x ) we need to
1
Determine the decomposition of C (A) into cells
2
Compute on each big cell the homogeneous polynomial of
degree m − n coinciding with TA (x ) .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SUMMARIZING
In order to compute TA (x ) we need to
1
Determine the decomposition of C (A) into cells
2
Compute on each big cell the homogeneous polynomial of
degree m − n coinciding with TA (x ) .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SUMMARIZING
In order to compute TA (x ) we need to
1
Determine the decomposition of C (A) into cells
2
Compute on each big cell the homogeneous polynomial of
degree m − n coinciding with TA (x ) .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SUMMARIZING
In order to compute TA (x ) we need to
1
Determine the decomposition of C (A) into cells
2
Compute on each big cell the homogeneous polynomial of
degree m − n coinciding with TA (x ) .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SUMMARIZING
In order to compute TA (x ) we need to
1
Determine the decomposition of C (A) into cells
2
Compute on each big cell the homogeneous polynomial of
degree m − n coinciding with TA (x ) .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
BOX SPLINES
While the function TA (x ) is the basic object, the more interesting
object for numerical analysis is the
box spline
that is the function BA (x ) characterized by the formula:
Z
Rn
m
X
Z
f (x )BA (x )dx =
f(
[0,1]m
ti ai )dt,
i=1
where f (x ) is any continuous function.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
BOX SPLINES
While the function TA (x ) is the basic object, the more interesting
object for numerical analysis is the
box spline
that is the function BA (x ) characterized by the formula:
Z
Rn
m
X
Z
f (x )BA (x )dx =
f(
[0,1]m
ti ai )dt,
i=1
where f (x ) is any continuous function.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
BOX SPLINES
While the function TA (x ) is the basic object, the more interesting
object for numerical analysis is the
box spline
that is the function BA (x ) characterized by the formula:
Z
Rn
m
X
Z
f (x )BA (x )dx =
f(
[0,1]m
ti ai )dt,
i=1
where f (x ) is any continuous function.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
THE BOX
The box spline BA (x ) is supported in the compact polytope:
The Box B(A)
that is the compact convex polytope
m
X
B(A) := {
ti ai }, 0 ≤ ti ≤ 1, ∀i.
i=1
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
THE BOX
The box spline BA (x ) is supported in the compact polytope:
The Box B(A)
that is the compact convex polytope
m
X
B(A) := {
ti ai }, 0 ≤ ti ≤ 1, ∀i.
i=1
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The box B(A) has a nice combinatorial structure, and can be
paved by a set of parallelepipeds indexed by:
all the bases which one can extract from A!
Example
In the next example
1 0 1
A=
0 1 1
−1 2 1
1 1 2
we have 15 bases and 15 parallelograms.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The box B(A) has a nice combinatorial structure, and can be
paved by a set of parallelepipeds indexed by:
all the bases which one can extract from A!
Example
In the next example
1 0 1
A=
0 1 1
−1 2 1
1 1 2
we have 15 bases and 15 parallelograms.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE
paving the box
1 0 1
A=
0 1 1
Claudio Procesi.
−1 2 1
1 1 2
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
1 0
=
0 1
START WITH
1 −1 2 1
1 1 1 2
1 0 A=
0 1 Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
1 0 1
A=
0 1 1














Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
1 0 1
A=
0 1 1




??
??
??
?
??
??
??
?
??
??
??
?





−1
1
??
??
??
?









Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
1 0 1
A=
0 1 1




??
??
??
?
??
??
??
?
−1 2
1 1
o
o ?
ooo ooooo ???
o
o
??
ooo
ooo
?
ooo oooo
o
??
??
o
o
??
??
oo
ooo
??
??
o
o
?
? ooo
o
o


ooo


o


o
o 


ooo 


ooo

ooo
ooooo

 o
oooo
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
•
o•
@@@
ooo o
o
@@
o
@@
ooo
o
o
oo
•
•
~~
~
~~ ~~ •
• •
•
ooo ooooo @@@
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• @o
• ~ • @@
o
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~~
ooo
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~ ~~
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•
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~
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~
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•
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~ ooooo
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~
~ oo
@@
~~ ooo
@@
~o~ooo
•
•
1 0 1 −1 2 1
A= 0 1 1 1 1 2
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
two box splines of class C 0 , h = 2 and C 1 , h = 3
1 0 1 A=
Hat function
0 1 1 or Courant element
Claudio Procesi.
1 0 1
A=
0 1 1
−1
1
Zwart-Powell element
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE OF A two dimensional box spline
1 2 1
Non continuous, A = 0 0 0
Claudio Procesi.
1 1
(h = 1)
0 2
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
3 reasons WHY the BOX SPLINE ?
recursive definition
Z
Rn
BA (x )dx = 1
Z 1
B[A,v ] (x ) =
0
BA (x − tv )dt
in the case of integral vectors, we have
PARTITION OF UNITY
The translates BA (x − λ), λ runs over the integral vectors form a
partition of 1.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
3 reasons WHY the BOX SPLINE ?
recursive definition
Z
Rn
BA (x )dx = 1
Z 1
B[A,v ] (x ) =
0
BA (x − tv )dt
in the case of integral vectors, we have
PARTITION OF UNITY
The translates BA (x − λ), λ runs over the integral vectors form a
partition of 1.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
3 reasons WHY the BOX SPLINE ?
recursive definition
Z
Rn
BA (x )dx = 1
Z 1
B[A,v ] (x ) =
0
BA (x − tv )dt
in the case of integral vectors, we have
PARTITION OF UNITY
The translates BA (x − λ), λ runs over the integral vectors form a
partition of 1.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
3 reasons WHY the BOX SPLINE ?
recursive definition
Z
Rn
BA (x )dx = 1
Z 1
B[A,v ] (x ) =
0
BA (x − tv )dt
in the case of integral vectors, we have
PARTITION OF UNITY
The translates BA (x − λ), λ runs over the integral vectors form a
partition of 1.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
TRIVIAL EXAMPLE
A = {1, 1} we have for the box spline
 ??
 ???

??


Now let us add to it its translates!
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
?
 ???

??

?

Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines




??  ??
?? ??
??
???
?


Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines




??
??
??
?
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines




??  ??
?? ??
??
???
?


Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines




??
??
??
?
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines




??  ??
?? ??
??
???
?


Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines




??
??
??
?
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines




??  ??
?? ??
??
???
?


Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
??
??
??
?




Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
??  ??
?? ??
??
???
?






Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
??
??
??
?




Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
?? 
??
???


?? 
??
???


Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
From TA to BA
For a given subset S of A define aS :=
P
a∈S
a
the basic formula is:
BA (x ) =
X
(−1)|S| TA (x − aS ).
S⊂A
So TA is the fundamental object.
Notice that the local pieces of BA are no more homogeneous
polynomials.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
From TA to BA
For a given subset S of A define aS :=
P
a∈S
a
the basic formula is:
BA (x ) =
X
(−1)|S| TA (x − aS ).
S⊂A
So TA is the fundamental object.
Notice that the local pieces of BA are no more homogeneous
polynomials.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
From TA to BA
For a given subset S of A define aS :=
P
a∈S
a
the basic formula is:
BA (x ) =
X
(−1)|S| TA (x − aS ).
S⊂A
So TA is the fundamental object.
Notice that the local pieces of BA are no more homogeneous
polynomials.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE OF THE HAT FUNCTION
1 0 1 A=
0 1 1 From the function TA
0
x
y
0
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE OF THE HAT FUNCTION
we get the box spline
translates of TA
hat function summing over the 6
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Here comes the algebra
How to compute TA ? or the partition function PA ?
We use the Laplace transform which will change the analytic
problem to one in
algebra
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
LAPLACE TRANSFORM from Rs = V to U = V ∗ .
Z
Lf (u) :=
e −hu | v i f (v )dv .
V
basic properties
p ∈ U, w ∈ V , write p, Dw for the linear function hp | v i and the
directional derivative on V
L(Dw f )(u) = wLf (u),
L(e p f )(u) = Lf (u − p),
Claudio Procesi.
L(pf )(u) = −Dp Lf (u),
L(f (v + w ))(u) = e w Lf (u).
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
LAPLACE TRANSFORM from Rs = V to U = V ∗ .
Z
Lf (u) :=
e −hu | v i f (v )dv .
V
basic properties
p ∈ U, w ∈ V , write p, Dw for the linear function hp | v i and the
directional derivative on V
L(Dw f )(u) = wLf (u),
L(e p f )(u) = Lf (u − p),
Claudio Procesi.
L(pf )(u) = −Dp Lf (u),
L(f (v + w ))(u) = e w Lf (u).
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
LAPLACE TRANSFORM from Rs = V to U = V ∗ .
Z
Lf (u) :=
e −hu | v i f (v )dv .
V
basic properties
p ∈ U, w ∈ V , write p, Dw for the linear function hp | v i and the
directional derivative on V
L(Dw f )(u) = wLf (u),
L(e p f )(u) = Lf (u − p),
Claudio Procesi.
L(pf )(u) = −Dp Lf (u),
L(f (v + w ))(u) = e w Lf (u).
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
LAPLACE TRANSFORM from Rs = V to U = V ∗ .
Z
Lf (u) :=
e −hu | v i f (v )dv .
V
basic properties
p ∈ U, w ∈ V , write p, Dw for the linear function hp | v i and the
directional derivative on V
L(Dw f )(u) = wLf (u),
L(e p f )(u) = Lf (u − p),
Claudio Procesi.
L(pf )(u) = −Dp Lf (u),
L(f (v + w ))(u) = e w Lf (u).
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Basic transformation rules
Consider any a ∈ V as a LINEAR FUNCTION ON U we have:
An easy computation gives the Laplace transforms:
LBA =
Y 1 − e −a
a∈A
a
and
LTA =
Y 1
a∈A
Claudio Procesi.
a
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Basic transformation rules
Consider any a ∈ V as a LINEAR FUNCTION ON U we have:
An easy computation gives the Laplace transforms:
LBA =
Y 1 − e −a
a∈A
a
and
LTA =
Y 1
a∈A
Claudio Procesi.
a
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Basic transformation rules
Consider any a ∈ V as a LINEAR FUNCTION ON U we have:
An easy computation gives the Laplace transforms:
LBA =
Y 1 − e −a
a∈A
a
and
LTA =
Y 1
a∈A
Claudio Procesi.
a
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
BASIC NON COMMUTATIVE ALGEBRAS
Algebraic Fourier transform
Weyl algebras
Set W (V ), W (U) be the two Weyl algebras of differential
operators with polynomial coefficients on V and U.
Weyl algebras
Set W (V ), W (U) be the two Weyl algebras of differential
operators with polynomial coefficients on V and U.
Fourier transform
There is an algebraic Fourier isomorphism between them, so any
W (V ) module M becomes a W (U) module M̂
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
BASIC NON COMMUTATIVE ALGEBRAS
Algebraic Fourier transform
Weyl algebras
Set W (V ), W (U) be the two Weyl algebras of differential
operators with polynomial coefficients on V and U.
Fourier transform
There is an algebraic Fourier isomorphism between them, so any
W (V ) module M becomes a W (U) module M̂
Fourier transform
There is an algebraic Fourier isomorphism between them, so any
W (V ) module M becomes a W (U) module M̂
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
BASIC NON COMMUTATIVE ALGEBRAS
Algebraic Fourier transform
Weyl algebras
Set W (V ), W (U) be the two Weyl algebras of differential
operators with polynomial coefficients on V and U.
Fourier transform
There is an algebraic Fourier isomorphism between them, so any
W (V ) module M becomes a W (U) module M̂
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
D−modules in Fourier duality:
Two modules Fourier isomorphic
1. The D−module DA := W (V )TA generated, in the space of
tempered distributions, by TA under the action of the algebra
W (V ) of differential operators on V with polynomial coefficients.
2. The algebra RA := S[V ][ a∈A a−1 ] obtained from the
Q
polynomials on U by inverting the element dA := a∈A a
Q
RA = W (U)dA−1
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
D−modules in Fourier duality:
Two modules Fourier isomorphic
1. The D−module DA := W (V )TA generated, in the space of
tempered distributions, by TA under the action of the algebra
W (V ) of differential operators on V with polynomial coefficients.
2. The algebra RA := S[V ][ a∈A a−1 ] obtained from the
Q
polynomials on U by inverting the element dA := a∈A a
Q
RA = W (U)dA−1
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Theorem
RA = W (U)dA−1
Under Laplace transform, L(TA ) = dA−1 and
D̂A is mapped isomorphically onto RA as W (U)-modules.
DA is the space of tempered distributions which are linear
combinations of polynomial functions on the cones C (B), B ⊂ A a
linearly independent subset and their distributional derivatives.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Theorem
RA = W (U)dA−1
Under Laplace transform, L(TA ) = dA−1 and
D̂A is mapped isomorphically onto RA as W (U)-modules.
DA is the space of tempered distributions which are linear
combinations of polynomial functions on the cones C (B), B ⊂ A a
linearly independent subset and their distributional derivatives.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Theorem
RA = W (U)dA−1
Under Laplace transform, L(TA ) = dA−1 and
D̂A is mapped isomorphically onto RA as W (U)-modules.
DA is the space of tempered distributions which are linear
combinations of polynomial functions on the cones C (B), B ⊂ A a
linearly independent subset and their distributional derivatives.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Theorem
RA = W (U)dA−1
Under Laplace transform, L(TA ) = dA−1 and
D̂A is mapped isomorphically onto RA as W (U)-modules.
DA is the space of tempered distributions which are linear
combinations of polynomial functions on the cones C (B), B ⊂ A a
linearly independent subset and their distributional derivatives.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Theorem
RA = W (U)dA−1
Under Laplace transform, L(TA ) = dA−1 and
D̂A is mapped isomorphically onto RA as W (U)-modules.
DA is the space of tempered distributions which are linear
combinations of polynomial functions on the cones C (B), B ⊂ A a
linearly independent subset and their distributional derivatives.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
RA and it is the coordinate ring of the open set AA
complement of the union of the hyperplanes of U
of equations a = 0, a ∈ A.
It is a cyclic module under W (U) generated by dA−1 .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
PROJECTIVE PICTURE OF THE ARRANGEMENT A3
We have drawn in the
projective plane of 4−tuples of
real numbers with sum 0, the
6 lines
xi − xj = 0, 1 ≤ i < j ≤ 4
the 7 intersection points are also subspaces of the arrangement!
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
PROJECTIVE PICTURE OF THE ARRANGEMENT A3
We have drawn in the
projective plane of 4−tuples of
real numbers with sum 0, the
6 lines
xi − xj = 0, 1 ≤ i < j ≤ 4
the 7 intersection points are also subspaces of the arrangement!
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The module RA
Localization
It is well known that once we invert an element in a polynomial
algebra we get a holonomic module over the algebra of differential
operators.
In particular it has a finite
composition series and it is cyclic
We want to describe a composition series of RA .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The module RA
Localization
It is well known that once we invert an element in a polynomial
algebra we get a holonomic module over the algebra of differential
operators.
In particular it has a finite
composition series and it is cyclic
We want to describe a composition series of RA .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The module RA
Localization
It is well known that once we invert an element in a polynomial
algebra we get a holonomic module over the algebra of differential
operators.
In particular it has a finite
composition series and it is cyclic
We want to describe a composition series of RA .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The building blocks
For each subspace W of U we have an irreducible module NW
(generated by the δ function of W ).
The NW as W runs over the subspaces of the hyperplane
arrangement given by the equations ai = 0, ai ∈ A are the
composition factors of RA .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Take coordinates x1 , . . . , xn
W = {x1 = x2 = . . . = xk = 0}
NW is generated by an element δ satisfying:
xi δ = 0, i ≤ k,
∂
δ, i > k.
∂xi
It is free generated by δ over:
C[x1 , x2 , . . . , xk ,
Claudio Procesi.
∂
∂
,...,
].
∂xk+1
∂xn
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Take coordinates x1 , . . . , xn
W = {x1 = x2 = . . . = xk = 0}
NW is generated by an element δ satisfying:
xi δ = 0, i ≤ k,
∂
δ, i > k.
∂xi
It is free generated by δ over:
C[x1 , x2 , . . . , xk ,
Claudio Procesi.
∂
∂
,...,
].
∂xk+1
∂xn
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The filtration of RA by polar order
Definition
Introduce a filtration by W (U)−submodules:
filtration degree ≤ k
RA,k is the span of all the fractions f a∈A a−ha , ha ≥ 0
for which the set of vectors a, with ha > 0, spans a space of
dimension ≤ k.
Q
We have RA,s = RA .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The filtration of RA by polar order
Definition
Introduce a filtration by W (U)−submodules:
filtration degree ≤ k
RA,k is the span of all the fractions f a∈A a−ha , ha ≥ 0
for which the set of vectors a, with ha > 0, spans a space of
dimension ≤ k.
Q
We have RA,s = RA .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The filtration of RA by polar order
RA,k /RA,k−1 is semisimple.
Its isotypic components are of type NW as W runs over the
subspaces of the arrangement of codimension k.
In particular RA,s /RA,s−1 is a direct sum of modules N0 .
The annihilator of the cyclic generator of N0 is generated by
all the variables, i.e. by all the elements ai ∈ A.
N0 is a free rank 1 module over the commutative polynomial
ring S[U] of differential operators with constant coefficients.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The filtration of RA by polar order
RA,k /RA,k−1 is semisimple.
Its isotypic components are of type NW as W runs over the
subspaces of the arrangement of codimension k.
In particular RA,s /RA,s−1 is a direct sum of modules N0 .
The annihilator of the cyclic generator of N0 is generated by
all the variables, i.e. by all the elements ai ∈ A.
N0 is a free rank 1 module over the commutative polynomial
ring S[U] of differential operators with constant coefficients.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The filtration of RA by polar order
RA,k /RA,k−1 is semisimple.
Its isotypic components are of type NW as W runs over the
subspaces of the arrangement of codimension k.
In particular RA,s /RA,s−1 is a direct sum of modules N0 .
The annihilator of the cyclic generator of N0 is generated by
all the variables, i.e. by all the elements ai ∈ A.
N0 is a free rank 1 module over the commutative polynomial
ring S[U] of differential operators with constant coefficients.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The filtration of RA by polar order
RA,k /RA,k−1 is semisimple.
Its isotypic components are of type NW as W runs over the
subspaces of the arrangement of codimension k.
In particular RA,s /RA,s−1 is a direct sum of modules N0 .
The annihilator of the cyclic generator of N0 is generated by
all the variables, i.e. by all the elements ai ∈ A.
N0 is a free rank 1 module over the commutative polynomial
ring S[U] of differential operators with constant coefficients.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The filtration of RA by polar order
RA,k /RA,k−1 is semisimple.
Its isotypic components are of type NW as W runs over the
subspaces of the arrangement of codimension k.
In particular RA,s /RA,s−1 is a direct sum of modules N0 .
The annihilator of the cyclic generator of N0 is generated by
all the variables, i.e. by all the elements ai ∈ A.
N0 is a free rank 1 module over the commutative polynomial
ring S[U] of differential operators with constant coefficients.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The filtration of RA by polar order
RA,k /RA,k−1 is semisimple.
Its isotypic components are of type NW as W runs over the
subspaces of the arrangement of codimension k.
In particular RA,s /RA,s−1 is a direct sum of modules N0 .
The annihilator of the cyclic generator of N0 is generated by
all the variables, i.e. by all the elements ai ∈ A.
N0 is a free rank 1 module over the commutative polynomial
ring S[U] of differential operators with constant coefficients.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A basis for RA,s /RA,s−1
By the previous theorem RA,s /RA,s−1 is a free module over
S[U] = C[
∂
∂
,...,
]
∂x1
∂xs
(in coordinates).
It is important to choose a basis.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
No broken circuits
A basis comes from the theory of matroids.
Let c := ai1 , . . . , aik ∈ A, i1 < i2 · · · < ik , be a sublist of linearly
independent elements.
Definition
We say that ai
that:
breaks c if there is an index 1 ≤ e ≤ k such
i ≤ ie .
ai is linearly dependent on
Claudio Procesi.
aie , . . . , aik .
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
No broken circuits
A basis comes from the theory of matroids.
Let c := ai1 , . . . , aik ∈ A, i1 < i2 · · · < ik , be a sublist of linearly
independent elements.
Definition
We say that ai
that:
breaks c if there is an index 1 ≤ e ≤ k such
i ≤ ie .
ai is linearly dependent on
Claudio Procesi.
aie , . . . , aik .
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The no broken bases extracted from A
no broken bases
For a basis b := ai1 , . . . , ais extracted from A set:
B(b) := {a ∈ A | a breaks b} and n(b) = |B(b)| the cardinality
of B(b).
Definition
We say that b is
no broken if B(b) = b or n(b) = s.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE of the elements braking a basis
Example A3 ordered as:
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 .
Let us draw in Red one particular basis and in Green the elements
which break it.
We have 16 bases
10 broken and 6 no broken
FIRST THE 10 broken
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE of the elements braking a basis
Example A3 ordered as:
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 .
Let us draw in Red one particular basis and in Green the elements
which break it.
We have 16 bases
10 broken and 6 no broken
FIRST THE 10 broken
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE of the elements braking a basis
Example A3 ordered as:
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 .
Let us draw in Red one particular basis and in Green the elements
which break it.
We have 16 bases
10 broken and 6 no broken
FIRST THE 10 broken
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE of the elements braking a basis
Example A3 ordered as:
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 .
Let us draw in Red one particular basis and in Green the elements
which break it.
We have 16 bases
10 broken and 6 no broken
FIRST THE 10 broken
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Example
A3 ordered as: α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 .
We have 6 n.b.b all contain necessarily α1 :
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 .
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 .
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 .
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 .
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 .
α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Let us visualize the simplices generated by the 6 n.b.b:
α21
1
111
11
11
11
11
11
11
11
11
11
11
11
α3
α1
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α2
999
99
99
99
99
99
9
α2 + α3
pp
ppp
p
p
pp
ppp
p
p
pp
ppp
pppp
p
α1
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α2
α1 + α2 + α3
kk
k
k
kk
k
k
kk
kkkkk
α1
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α1 + αL2
α1
LLL
LLL
LLL
LLL
LLL
LLL
LLL
Claudio Procesi.
α3
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α + α2 + Sα3
α1
1
kkk
k
k
k
kkk
kkk
k
k
k
Claudio Procesi.
SSS
SSS
SSS
SSS
SS
α3
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
α1 + α2
α2 + α3
mm
mmm
m
m
mmm
mmm
m
m
mmm
mmmmm
m
mmmm
α1
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Decomposition into big cells and no broken bases
Let us visualize the decomposition into big cells, obtained
overlapping the cones generated by no broken bases.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A remarkable fact
//
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//
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///
//
//
//
//
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A remarkable fact
//
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t
tt //
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tt
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tttt
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ttt
t
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A remarkable fact
//
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tt //
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Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A remarkable fact
//
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JJJ
t
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tt //
t
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///
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ttttJJJJ
t
t
t
/
t
J
t
t
J
t
JJ //
tt
JJ /
tttttttt
JJ /
tt
JJ/
t tt
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A remarkable fact
//
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JJJ
t
JJ
tt //
t
JJJtt
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tttttJJJJJJJ
t
t
/
J
t
t
J
J
t
t
J
JJJJ //
tt
JJJJ /
tttttttttttt
JJJJ /
ttt
JJJJ/
tttttt
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A remarkable fact
//
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JJJ
t
JJ
tt //
t
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ttttJJJJJJJ
t
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/
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t
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t
J
JJJJ //
tt
JJJJ /
tttttttttttt
JJJJ /
ttt
JJJJ/
tttttt
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A remarkable fact
//
///
//
//
///
///
///
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JJJ
t
JJ
tt //
t
JJJtt
///
//
ttttJJJJJJJ
t
t
t
/
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t
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J
t
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JJJJ //
tt
JJJJ /
tttttttttttt
JJJJ /
ttt
JJJJ/
tttttt
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The overlapping theorem
You have visually seen a theorem we proved in general:
by overlapping the cones generated by the no broken bases one
obtains the entire decomposition into big cells!!
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The basis theorem
Theorem
A basis for RA,s /RA,s−1 over S[U] is given by
the classes of the elements a∈b a−1 as b runs
over the set of no broken bases.
Q
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The expansion of dA−1
Denote by N B the no broken bases extracted from A.
We have a more precise THEOREM.
dA−1 =
X
b∈N B
pb
Y
a−1 , pb ∈ S[U] = C[
a∈b
Claudio Procesi.
∂
∂
,...,
]
∂x1
∂xs
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE Courant element
Example
USE COORDINATES x , y SET
1 0
A = [x + y , x , y ] = [x , y ] 1 1
1
0
1
1
1
=
+
=
2
(x + y ) x y
x (x + y )
y (x + y )2
−
∂
1
∂
1
−
∂y x (x + y )
∂x y (x + y )
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE ZP element
Example
1 0
A = [x + y , x , y , −x + y ] = [x , y ] 1 1
1 −1
0 1
1
1
4
1
=
−
+
=
3
3
(x + y ) x y (−x + y )
x (x + y ) (x + y ) (−x + y ) y (x + y )3
1/2[
∂2
1
∂
1
1
∂2
∂ 2
+(
− 2
]
+
)
2
∂ y x (x + y )
∂x ∂y
(x + y )(−x + y ) ∂ x y (x + y )
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
We now need the basic inversion
Let X = {a1 , . . . , an } be a basis,d := | det(a1 , . . . , as )|
χC (X ) the characteristic function of the positive quadrant C (X )
generated by X .
Basic example of inversion
L(d −1 χC (X ) ) =
n
Y
ai−1
i=1
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
We now need the basic inversion
Let X = {a1 , . . . , an } be a basis,d := | det(a1 , . . . , as )|
χC (X ) the characteristic function of the positive quadrant C (X )
generated by X .
Basic example of inversion
L(d −1 χC (X ) ) =
n
Y
ai−1
i=1
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
We now need the basic inversion
Let X = {a1 , . . . , an } be a basis,d := | det(a1 , . . . , as )|
χC (X ) the characteristic function of the positive quadrant C (X )
generated by X .
Basic example of inversion
L(d −1 χC (X ) ) =
n
Y
ai−1
i=1
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
We now need the basic inversion
Let X = {a1 , . . . , an } be a basis,d := | det(a1 , . . . , as )|
χC (X ) the characteristic function of the positive quadrant C (X )
generated by X .
Basic example of inversion
L(d −1 χC (X ) ) =
n
Y
ai−1
i=1
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
We are ready to invert!
We want to invert
dA−1 =
X
pb,A (
b∈N B
∂
∂ Y −1
a
,...,
)
∂x1
∂xs a∈b
From the basic example and the properties!
We get
X
pb,A (−x1 , . . . , −xs )db−1 χC (b)
b∈N B
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
We are ready to invert!
We want to invert
dA−1 =
X
pb,A (
b∈N B
∂
∂ Y −1
a
,...,
)
∂x1
∂xs a∈b
From the basic example and the properties!
We get
X
pb,A (−x1 , . . . , −xs )db−1 χC (b)
b∈N B
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
EXAMPLE ZP element
Inverting
1/2[
∂2
∂2
1
∂
1
1
∂
+( + )2
− 2
]
2
∂ y x (x + y )
∂x ∂y
(x + y )(−x + y ) ∂ x y (x + y )
we get
1/2[y 2 χC ((1,0),(1,1)) +
(x + y )2
χC ((1,1),(−1,1) − x 2 χC ((0,1),(1,1)) ]
2
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The C 1 function TA , case ZP
1 1 0
A=
1 0 1
−1
1
_??
O
?
??


?? 
? 
/
2TA is 0 outside the cone and on the three cells:
2
(x +y )
??
2
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
Claudio Procesi.
0
?
(x +y )2
2
}
}}
}
}}
}}
}
}}
}}
−x 2
}
}}
}}
}
}
}}
}
}}
}}
}
}}
}}
}
Topics in Hyperplane
Arrangements,
}
y 2 Polytopes and Box-Splines
The C 1 function TA , case ZP
the associated cone C (A) has three big cells
2TA is 0
??
??


??

??


??

??

??

??

??


??

??


??

??

?_ ?
O
?
??

?? 
? 
/
outside the cone and on the three cells:
??
??
??
??
??
??
(x +y )2
2
(x +y )2
2
Claudio Procesi.
}}
}}
}
}
}}
}
}
Topics in Hyperplane Arrangements, Polytopes
} and Box-Splines
The C 1 function TA , case ZP
2TA is 0 outside the cone and on the three cells:
2
2
(x +y )
(x +y )
??
2
2
}}
??
}}
??
}
??
}}
??
}}
}
??
}}
??
−x 2
}}
??
}
}
??
}}
??
}
}
??
}
??
}}
}
??
}}
??
}}
??
}
??
}}
??
}}
}
y2
0
??
}
}
??
}
??
}}
?? }}}
}
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
GENERAL FORMULA
Given a point x in the closure of a big cell c we have
Jeffry-Kirwan residue formula
TA (x ) =
X
| det(b)|−1 pb,A (−x ).
b | c⊂C (b)
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
I know of three methods to compute the polynomials pb,A (−x ).
1
Develop explicitly dA−1 by a recursive algorithm.
2
Use a suitable system of differential equations which are
interpreted as linear equations on the polynomials.
3
Compute them as
residues at suitable points at infinity.
PROBLEM: Compare the three methods from the point of view of
efficiency of the algorithm.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
I know of three methods to compute the polynomials pb,A (−x ).
1
Develop explicitly dA−1 by a recursive algorithm.
2
Use a suitable system of differential equations which are
interpreted as linear equations on the polynomials.
3
Compute them as
residues at suitable points at infinity.
PROBLEM: Compare the three methods from the point of view of
efficiency of the algorithm.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
I know of three methods to compute the polynomials pb,A (−x ).
1
Develop explicitly dA−1 by a recursive algorithm.
2
Use a suitable system of differential equations which are
interpreted as linear equations on the polynomials.
3
Compute them as
residues at suitable points at infinity.
PROBLEM: Compare the three methods from the point of view of
efficiency of the algorithm.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
I know of three methods to compute the polynomials pb,A (−x ).
1
Develop explicitly dA−1 by a recursive algorithm.
2
Use a suitable system of differential equations which are
interpreted as linear equations on the polynomials.
3
Compute them as
residues at suitable points at infinity.
PROBLEM: Compare the three methods from the point of view of
efficiency of the algorithm.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
I know of three methods to compute the polynomials pb,A (−x ).
1
Develop explicitly dA−1 by a recursive algorithm.
2
Use a suitable system of differential equations which are
interpreted as linear equations on the polynomials.
3
Compute them as
residues at suitable points at infinity.
PROBLEM: Compare the three methods from the point of view of
efficiency of the algorithm.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
I know of three methods to compute the polynomials pb,A (−x ).
1
Develop explicitly dA−1 by a recursive algorithm.
2
Use a suitable system of differential equations which are
interpreted as linear equations on the polynomials.
3
Compute them as
residues at suitable points at infinity.
PROBLEM: Compare the three methods from the point of view of
efficiency of the algorithm.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The differential equations
We need a basic definition of combinatorial nature
Definition
We say that a sublist Y ⊂ A is a
A − Y do not span V .
cocircuit, if the elements in
The basic differential operators
Q
For such Y set DY := a∈Y Da , a differential operator with
constant coefficients.
(Da is directional derivative, first order operator)
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The differential equations
We need a basic definition of combinatorial nature
Definition
We say that a sublist Y ⊂ A is a
A − Y do not span V .
cocircuit, if the elements in
The basic differential operators
Q
For such Y set DY := a∈Y Da , a differential operator with
constant coefficients.
(Da is directional derivative, first order operator)
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
For a given no broken circuit basis b, consider the element
Q
Db := a∈b
/ Da .
Characterization by differential equations
The polynomials pb,A are characterized by the differential equations
DY p = 0, ∀Y ,
a cocircuit in A
(
Db pc,A (x1 , . . . , xs ) =
Claudio Procesi.
1
0
if
if
b=c
b 6= c
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
For a given no broken circuit basis b, consider the element
Q
Db := a∈b
/ Da .
Characterization by differential equations
The polynomials pb,A are characterized by the differential equations
DY p = 0, ∀Y ,
a cocircuit in A
(
Db pc,A (x1 , . . . , xs ) =
Claudio Procesi.
1
0
if
if
b=c
b 6= c
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
For a given no broken circuit basis b, consider the element
Q
Db := a∈b
/ Da .
Characterization by differential equations
The polynomials pb,A are characterized by the differential equations
DY p = 0, ∀Y ,
a cocircuit in A
(
Db pc,A (x1 , . . . , xs ) =
Claudio Procesi.
1
0
if
if
b=c
b 6= c
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The space D(A)
A remarkable space of polynomials
D(A) := {p | DY p = 0, ∀Y ,
Claudio Procesi.
a cocircuit in A}
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The theorem of Dhamen Micchelli
dim D(A) equals the total number of bases extracted from A
Recall the definitions
B(A) denotes the set of bases extracted from A.
n(b) is the number of elements of A breaking b.
We have a more precise theorem
The polynomials pb,A form a basis for the top
degree part (m − n) of D(A).
The graded dimension of D(A) is given by
X
HA (q) =
q m−n(b) .
b∈B(A)
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The theorem of Dhamen Micchelli
dim D(A) equals the total number of bases extracted from A
Recall the definitions
B(A) denotes the set of bases extracted from A.
n(b) is the number of elements of A breaking b.
We have a more precise theorem
The polynomials pb,A form a basis for the top
degree part (m − n) of D(A).
The graded dimension of D(A) is given by
X
HA (q) =
q m−n(b) .
b∈B(A)
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The theorem of Dhamen Micchelli
dim D(A) equals the total number of bases extracted from A
Recall the definitions
B(A) denotes the set of bases extracted from A.
n(b) is the number of elements of A breaking b.
We have a more precise theorem
The polynomials pb,A form a basis for the top
degree part (m − n) of D(A).
The graded dimension of D(A) is given by
X
HA (q) =
q m−n(b) .
b∈B(A)
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The theorem of Dhamen Micchelli
dim D(A) equals the total number of bases extracted from A
Recall the definitions
B(A) denotes the set of bases extracted from A.
n(b) is the number of elements of A breaking b.
We have a more precise theorem
The polynomials pb,A form a basis for the top
degree part (m − n) of D(A).
The graded dimension of D(A) is given by
X
HA (q) =
q m−n(b) .
b∈B(A)
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The theorem of Dhamen Micchelli
dim D(A) equals the total number of bases extracted from A
Recall the definitions
B(A) denotes the set of bases extracted from A.
n(b) is the number of elements of A breaking b.
We have a more precise theorem
The polynomials pb,A form a basis for the top
degree part (m − n) of D(A).
The graded dimension of D(A) is given by
X
HA (q) =
q m−n(b) .
b∈B(A)
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
For A3 we get
The graded dimension is:
6q 3 + 6q 2 + 3q + 1
Remark that for all polynomials in three variables it is:
. . . + 10q 3 + 6q 2 + 3q + 1
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A commutative algebra
By standard duality the previous theorems amount to study the
quotient of the symmetric algebra S[V ] by the graded ideal
IA
generated by all products
Q
a∈Y
a, Y ∈ E(A)
as Y varies over the cocircuits.
BASIC THEOREM
The graded algebra S[V ]/IA is finite dimensional of graded
dimension:
X
HA (q) =
q m−n(b) .
b∈B(A)
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A commutative algebra
By standard duality the previous theorems amount to study the
quotient of the symmetric algebra S[V ] by the graded ideal
IA
generated by all products
Q
a∈Y
a, Y ∈ E(A)
as Y varies over the cocircuits.
BASIC THEOREM
The graded algebra S[V ]/IA is finite dimensional of graded
dimension:
X
HA (q) =
q m−n(b) .
b∈B(A)
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Connection with non commutative object
IA is the annihilator ideal of the class of dA−1 in the W (V ) module
RA,n /RA,n−1 thought of just as S[V ] module.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
THE DISCRETE CASE
We pass to integral vectors and the partition function.
Denote by Λ the integral lattice.
It is convenient to think of a function f on Λ as the
distribution
X
f (λ)δλ .
with Laplace transform
λ∈Λ
X
f (λ)e −λ .
λ∈Λ
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
THE DISCRETE CASE
We pass to integral vectors and the partition function.
Denote by Λ the integral lattice.
It is convenient to think of a function f on Λ as the
distribution
X
f (λ)δλ .
with Laplace transform
λ∈Λ
X
f (λ)e −λ .
λ∈Λ
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
THE DISCRETE CASE
We pass to integral vectors and the partition function.
Denote by Λ the integral lattice.
It is convenient to think of a function f on Λ as the
distribution
X
f (λ)δλ .
with Laplace transform
λ∈Λ
X
f (λ)e −λ .
λ∈Λ
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The partition function
In particular for the partition function
PA (b) = #{t1 , . . . , tm ∈ N |
m
X
ti ai = b}
i=1
An easy computation gives its Laplace transform:
For the partition
distribution
PA =
X
we have
PA (b)δb
LPA =
1
(1 − e −a )
a∈A
Y
λ∈Λ
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The partition function
In particular for the partition function
PA (b) = #{t1 , . . . , tm ∈ N |
m
X
ti ai = b}
i=1
An easy computation gives its Laplace transform:
For the partition
distribution
PA =
X
we have
PA (b)δb
LPA =
1
(1 − e −a )
a∈A
Y
λ∈Λ
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
THE FINAL FORMULA
There is a parallel theory, as a result we can compute a set of
polynomials qb,φ (−x ) indexed by pairs, a character φ of finite
order and a no broken basis in Aφ = {a ∈ A | φ(e a ) = 1}.
The analogue of the Jeffrey–Kirwan formula is:
Theorem
Given a point x in the closure of a big cell c we have
Residue formula for partition function
PA (x ) =
X
φ∈P(A)
X
eφ
qb,φ (−x )
b∈N BAφ | c⊂C (b)
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
THE FINAL FORMULA
There is a parallel theory, as a result we can compute a set of
polynomials qb,φ (−x ) indexed by pairs, a character φ of finite
order and a no broken basis in Aφ = {a ∈ A | φ(e a ) = 1}.
The analogue of the Jeffrey–Kirwan formula is:
Theorem
Given a point x in the closure of a big cell c we have
Residue formula for partition function
PA (x ) =
X
φ∈P(A)
X
eφ
qb,φ (−x )
b∈N BAφ | c⊂C (b)
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
OTHER BASIC NON COMMUTATIVE ALGEBRAS
The periodic Weyl algebras W̃ (U) and W # (Λ)
W̃ (U) the algebra of differential operators on U with coefficients
exponential functions e λ , λ ∈ Λ.
W # (Λ) the algebra of difference operators with polynomial
coefficients on Λ.
Fourier transform
There is an algebraic Fourier isomorphism between W̃ (U) and
W # (Λ), so any W̃ (U) module M becomes a W # (Λ) module M̂
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
OTHER BASIC NON COMMUTATIVE ALGEBRAS
The periodic Weyl algebras W̃ (U) and W # (Λ)
W̃ (U) the algebra of differential operators on U with coefficients
exponential functions e λ , λ ∈ Λ.
W # (Λ) the algebra of difference operators with polynomial
coefficients on Λ.
Fourier transform
There is an algebraic Fourier isomorphism between W̃ (U) and
W # (Λ), so any W̃ (U) module M becomes a W # (Λ) module M̂
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
OTHER BASIC NON COMMUTATIVE ALGEBRAS
The periodic Weyl algebras W̃ (U) and W # (Λ)
W̃ (U) the algebra of differential operators on U with coefficients
exponential functions e λ , λ ∈ Λ.
W # (Λ) the algebra of difference operators with polynomial
coefficients on Λ.
Fourier transform
There is an algebraic Fourier isomorphism between W̃ (U) and
W # (Λ), so any W̃ (U) module M becomes a W # (Λ) module M̂
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
D−modules in Fourier duality:
Two modules Fourier isomorphic
1. The W # −module DA# := W # (Λ)PA generated, in the space of
tempered distributions, by the partition distribution PA .
2. The W (U) module SA = W̃ (U)uA−1 generated in the space of
Q
functions by uA−1 := a∈A (1 − e −a )−1 .
Fact!
SA = C[Λ][ a∈A (1 − e −a )−1 ] is the algebra obtained from the
Q
character ring C[Λ] by inverting uA := a∈A (1 − e −a ).
Q
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
D−modules in Fourier duality:
Two modules Fourier isomorphic
1. The W # −module DA# := W # (Λ)PA generated, in the space of
tempered distributions, by the partition distribution PA .
2. The W (U) module SA = W̃ (U)uA−1 generated in the space of
Q
functions by uA−1 := a∈A (1 − e −a )−1 .
Fact!
SA = C[Λ][ a∈A (1 − e −a )−1 ] is the algebra obtained from the
Q
character ring C[Λ] by inverting uA := a∈A (1 − e −a ).
Q
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
D−modules in Fourier duality:
Two modules Fourier isomorphic
1. The W # −module DA# := W # (Λ)PA generated, in the space of
tempered distributions, by the partition distribution PA .
2. The W (U) module SA = W̃ (U)uA−1 generated in the space of
Q
functions by uA−1 := a∈A (1 − e −a )−1 .
Fact!
SA = C[Λ][ a∈A (1 − e −a )−1 ] is the algebra obtained from the
Q
character ring C[Λ] by inverting uA := a∈A (1 − e −a ).
Q
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Theorem
SA = W̃ (U)uA−1
Under Laplace transform, L(PA ) = uA−1 and
D̂A# is mapped isomorphically onto SA as W̃ (U)-modules.
DA# is the space of tempered distributions which are linear
combinations of polynomial functions on the cones C (B) ∩ Λ,
B ⊂ A a linearly independent subset and their distributional
derivatives.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Theorem
SA = W̃ (U)uA−1
Under Laplace transform, L(PA ) = uA−1 and
D̂A# is mapped isomorphically onto SA as W̃ (U)-modules.
DA# is the space of tempered distributions which are linear
combinations of polynomial functions on the cones C (B) ∩ Λ,
B ⊂ A a linearly independent subset and their distributional
derivatives.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Theorem
SA = W̃ (U)uA−1
Under Laplace transform, L(PA ) = uA−1 and
D̂A# is mapped isomorphically onto SA as W̃ (U)-modules.
DA# is the space of tempered distributions which are linear
combinations of polynomial functions on the cones C (B) ∩ Λ,
B ⊂ A a linearly independent subset and their distributional
derivatives.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Theorem
SA = W̃ (U)uA−1
Under Laplace transform, L(PA ) = uA−1 and
D̂A# is mapped isomorphically onto SA as W̃ (U)-modules.
DA# is the space of tempered distributions which are linear
combinations of polynomial functions on the cones C (B) ∩ Λ,
B ⊂ A a linearly independent subset and their distributional
derivatives.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Theorem
SA = W̃ (U)uA−1
Under Laplace transform, L(PA ) = uA−1 and
D̂A# is mapped isomorphically onto SA as W̃ (U)-modules.
DA# is the space of tempered distributions which are linear
combinations of polynomial functions on the cones C (B) ∩ Λ,
B ⊂ A a linearly independent subset and their distributional
derivatives.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The toric arrangement
The toric arrangement
T the torus of character group Λ
SA = C[Λ][uA−1 ] is the coordinate ring of the open set TA ⊂ T
complement of the union of the subgroups of T of equations
e a = 1, a ∈ A.
SA is a cyclic module under W # (U) generated by uA−1 .
The elements of the toric arrangement
are the connected components of all the intersections of the
subgroups e a = 1, a ∈ A.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The toric arrangement
The toric arrangement
T the torus of character group Λ
SA = C[Λ][uA−1 ] is the coordinate ring of the open set TA ⊂ T
complement of the union of the subgroups of T of equations
e a = 1, a ∈ A.
SA is a cyclic module under W # (U) generated by uA−1 .
The elements of the toric arrangement
are the connected components of all the intersections of the
subgroups e a = 1, a ∈ A.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The toric arrangement
The toric arrangement
T the torus of character group Λ
SA = C[Λ][uA−1 ] is the coordinate ring of the open set TA ⊂ T
complement of the union of the subgroups of T of equations
e a = 1, a ∈ A.
SA is a cyclic module under W # (U) generated by uA−1 .
The elements of the toric arrangement
are the connected components of all the intersections of the
subgroups e a = 1, a ∈ A.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The toric arrangement
The toric arrangement
T the torus of character group Λ
SA = C[Λ][uA−1 ] is the coordinate ring of the open set TA ⊂ T
complement of the union of the subgroups of T of equations
e a = 1, a ∈ A.
SA is a cyclic module under W # (U) generated by uA−1 .
The elements of the toric arrangement
are the connected components of all the intersections of the
subgroups e a = 1, a ∈ A.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Given a basis b extracted from A, consider the lattice Λb ⊂ Λ that
it generates in Λ.
We have that Λ/Λb is a finite group of order [Λ : Λb ] = | det(b)|.
Its character group is the finite subgroup T (b) of T which is the
intersection of the kernels of the functions e a as a ∈ b.
We now define the
points of the arrangement
P(A) := ∪b∈B(A) T (b).
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Given a basis b extracted from A, consider the lattice Λb ⊂ Λ that
it generates in Λ.
We have that Λ/Λb is a finite group of order [Λ : Λb ] = | det(b)|.
Its character group is the finite subgroup T (b) of T which is the
intersection of the kernels of the functions e a as a ∈ b.
We now define the
points of the arrangement
P(A) := ∪b∈B(A) T (b).
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Given a basis b extracted from A, consider the lattice Λb ⊂ Λ that
it generates in Λ.
We have that Λ/Λb is a finite group of order [Λ : Λb ] = | det(b)|.
Its character group is the finite subgroup T (b) of T which is the
intersection of the kernels of the functions e a as a ∈ b.
We now define the
points of the arrangement
P(A) := ∪b∈B(A) T (b).
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Given a basis b extracted from A, consider the lattice Λb ⊂ Λ that
it generates in Λ.
We have that Λ/Λb is a finite group of order [Λ : Λb ] = | det(b)|.
Its character group is the finite subgroup T (b) of T which is the
intersection of the kernels of the functions e a as a ∈ b.
We now define the
points of the arrangement
P(A) := ∪b∈B(A) T (b).
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
THE FILTRATION
We have as for hyperplanes a filtration by polar orders on SA .
Each graded piece is semisimple.
The isopypic components appearing in grade k correspond to the
connected components of the toric arrangement of codimension k
For the top part SA,n /SA,n−1
we have a sum over the
points of the arrangement P(A).
The isotypic component associated to a point e φ decomposes as
direct sum of irreducibles indexed by the no broken bases in
Aφ := {a ∈ A | e ha | φi = 1}.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
THE FILTRATION
We have as for hyperplanes a filtration by polar orders on SA .
Each graded piece is semisimple.
The isopypic components appearing in grade k correspond to the
connected components of the toric arrangement of codimension k
For the top part SA,n /SA,n−1
we have a sum over the
points of the arrangement P(A).
The isotypic component associated to a point e φ decomposes as
direct sum of irreducibles indexed by the no broken bases in
Aφ := {a ∈ A | e ha | φi = 1}.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
THE FILTRATION
We have as for hyperplanes a filtration by polar orders on SA .
Each graded piece is semisimple.
The isopypic components appearing in grade k correspond to the
connected components of the toric arrangement of codimension k
For the top part SA,n /SA,n−1
we have a sum over the
points of the arrangement P(A).
The isotypic component associated to a point e φ decomposes as
direct sum of irreducibles indexed by the no broken bases in
Aφ := {a ∈ A | e ha | φi = 1}.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
THE FILTRATION
We have as for hyperplanes a filtration by polar orders on SA .
Each graded piece is semisimple.
The isopypic components appearing in grade k correspond to the
connected components of the toric arrangement of codimension k
For the top part SA,n /SA,n−1
we have a sum over the
points of the arrangement P(A).
The isotypic component associated to a point e φ decomposes as
direct sum of irreducibles indexed by the no broken bases in
Aφ := {a ∈ A | e ha | φi = 1}.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The previous formula shows in particular, that the partition
function is on each cell a
Local structure of PA
LINEAR COMBINATION OF
POLYNOMIALS TIME
PERIODIC EXPONENTIALS.
Claudio Procesi.
Such a function is called a
QUASI POLINOMIAL
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Basic equations
As for the case of the multivariate spline:
The quasi polynomials appearing in the formula for PA satisfy
special difference equations
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
DIFFERENCE OPERATORS
For a ∈ Λ and f a function on Λ we define the the
difference operator:
∇a f (x ) = f (x ) − f (x − a), ∇a = 1 − τa .
Example
As special functions we have the characters, eigenvectors of
difference operators.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
DIFFERENCE EQUATIONS
Parallel to the study of D(A), one can study the
system of difference equations
∇Y f = 0, where ∇Y :=
Q
v ∈Y
∇v
as Y ∈ E(A) runs over the cocircuits.
Let us denote the space of solutions by:
∇(A) := {f : Λ → C, | ∇Y (f ) = 0, ∀Y ∈ E(A)}.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
DIFFERENCE EQUATIONS
Parallel to the study of D(A), one can study the
system of difference equations
∇Y f = 0, where ∇Y :=
Q
v ∈Y
∇v
as Y ∈ E(A) runs over the cocircuits.
Let us denote the space of solutions by:
∇(A) := {f : Λ → C, | ∇Y (f ) = 0, ∀Y ∈ E(A)}.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The equations ∇Y f = 0 via commutative algebra.
The ideal JA
The products NY := v ∈Y (1 − e −v ) as Y runs
over the cocircuits in A, generate an ideal
JA ⊂ C[Λ]
Q
Duality
The space of functions ∇(A), which we want to
describe, is the dual of C[Λ]/JA .
Finite dimension
This space is finite dimensional its dimension is a
weighted analogue δ(A) of d(A) (number
of elements of B(A) the bases extracted from A).
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The equations ∇Y f = 0 via commutative algebra.
The ideal JA
The products NY := v ∈Y (1 − e −v ) as Y runs
over the cocircuits in A, generate an ideal
JA ⊂ C[Λ]
Q
Duality
The space of functions ∇(A), which we want to
describe, is the dual of C[Λ]/JA .
Finite dimension
This space is finite dimensional its dimension is a
weighted analogue δ(A) of d(A) (number
of elements of B(A) the bases extracted from A).
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
The equations ∇Y f = 0 via commutative algebra.
The ideal JA
The products NY := v ∈Y (1 − e −v ) as Y runs
over the cocircuits in A, generate an ideal
JA ⊂ C[Λ]
Q
Duality
The space of functions ∇(A), which we want to
describe, is the dual of C[Λ]/JA .
Finite dimension
This space is finite dimensional its dimension is a
weighted analogue δ(A) of d(A) (number
of elements of B(A) the bases extracted from A).
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Weighted dimension
X
δ(A) :=
| det(b)|.
b∈B(A)
This formula has a strict connection with the paving of the box.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Example
Let us take
0 1 1
A=
1 0 1
−1
1
See that
δ(A) = 1 + 1 + 1 + 1 + 1 + 2 = 7
is the number of points in
which the box B(A), shifted
generically a little, intersects
the lattice!
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Lemma
The variety defined by the ideal JA is
P(A) the finite set of points of the arrangement.
Now by elementary commutative algebra we know that
C[Λ]/JA = ⊕e φ ∈P(A) C[Λ]/JA (φ)
where C[Λ]/JA (φ) is its localization at e φ .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
FROM DIFFERENCE TO DIFFERENTIAL
EQUATIONS
logarithm isomorphism
There is a formal machinery which allows us to interpret, locally
around a point, difference equations as restriction to the lattice of
differential equations, we call it the
logarithm isomorphism
Claudio Procesi.
We have this for any module
over the periodic Weyl algebra
∂
C[ ∂x
, e xi ] as soon as for
i
algebraic reasons (nilpotency)
we can deduce from the action
of e xi also an action of xi .
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Recall we had the ideal IA spanned by
cocircuits.
Q
a∈Y
a, Y ∈ E(A)
Theorem
Under the logarithm isomorphism C[Λ]/JA (φ) becomes isomorphic
to the ring RAφ := S[V ]/IAφ . Thus:
C[Λ]/JA ∼
= ⊕e φ ∈P(A) RAφ .
In particular dim(C[Λ]/JA ) = δ(A).
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
From volumes to partition functions
Theorem
Given a point x in the closure of a big cell c we have
PA (x ) =
X
φ∈P(A)
Qφ =
X
b eφ
Q
φ
Y
a∈A
/ φ
pb,Aφ (−x ) .
b∈N BAφ | c⊂C (b)
Y
1
a − hφ | ai
−a
1 − e a∈A 1 − e −a+hφ | ai
Claudio Procesi.
φ
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SUMMARIZING
Given a list of vectors A.
We have the two functions TA , BA .
TA is supported on the cone C (A) and coincides on each big
cell with a homogeneous polynomial in the space D(A)
defined by the differential equations DY f = 0, Y ∈ E(A).
The space D(A) has as dimension the number d(A) of bases
extracted from A.
There are explicit formulas to compute TA on each cell.
BA is deduced from TA .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SUMMARIZING
Given a list of vectors A.
We have the two functions TA , BA .
TA is supported on the cone C (A) and coincides on each big
cell with a homogeneous polynomial in the space D(A)
defined by the differential equations DY f = 0, Y ∈ E(A).
The space D(A) has as dimension the number d(A) of bases
extracted from A.
There are explicit formulas to compute TA on each cell.
BA is deduced from TA .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SUMMARIZING
Given a list of vectors A.
We have the two functions TA , BA .
TA is supported on the cone C (A) and coincides on each big
cell with a homogeneous polynomial in the space D(A)
defined by the differential equations DY f = 0, Y ∈ E(A).
The space D(A) has as dimension the number d(A) of bases
extracted from A.
There are explicit formulas to compute TA on each cell.
BA is deduced from TA .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SUMMARIZING
Given a list of vectors A.
We have the two functions TA , BA .
TA is supported on the cone C (A) and coincides on each big
cell with a homogeneous polynomial in the space D(A)
defined by the differential equations DY f = 0, Y ∈ E(A).
The space D(A) has as dimension the number d(A) of bases
extracted from A.
There are explicit formulas to compute TA on each cell.
BA is deduced from TA .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
SUMMARIZING
Given a list of vectors A.
We have the two functions TA , BA .
TA is supported on the cone C (A) and coincides on each big
cell with a homogeneous polynomial in the space D(A)
defined by the differential equations DY f = 0, Y ∈ E(A).
The space D(A) has as dimension the number d(A) of bases
extracted from A.
There are explicit formulas to compute TA on each cell.
BA is deduced from TA .
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
When A is in a lattice we have the partition function PA .
PA is supported on the intersection of the lattice with the
cone C (A) and coincides on each big cell with a quasi
polynomial in the space ∇(A) defined by the difference
equations ∇Y f = 0, Y ∈ E(A).
The space ∇(A) has as dimension δ(A) the weighted number
of bases extracted from A.
There are explicit formulas to compute PA on each cell.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
When A is in a lattice we have the partition function PA .
PA is supported on the intersection of the lattice with the
cone C (A) and coincides on each big cell with a quasi
polynomial in the space ∇(A) defined by the difference
equations ∇Y f = 0, Y ∈ E(A).
The space ∇(A) has as dimension δ(A) the weighted number
of bases extracted from A.
There are explicit formulas to compute PA on each cell.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
When A is in a lattice we have the partition function PA .
PA is supported on the intersection of the lattice with the
cone C (A) and coincides on each big cell with a quasi
polynomial in the space ∇(A) defined by the difference
equations ∇Y f = 0, Y ∈ E(A).
The space ∇(A) has as dimension δ(A) the weighted number
of bases extracted from A.
There are explicit formulas to compute PA on each cell.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
When A is in a lattice we have the partition function PA .
PA is supported on the intersection of the lattice with the
cone C (A) and coincides on each big cell with a quasi
polynomial in the space ∇(A) defined by the difference
equations ∇Y f = 0, Y ∈ E(A).
The space ∇(A) has as dimension δ(A) the weighted number
of bases extracted from A.
There are explicit formulas to compute PA on each cell.
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
We have shown that there are interesting constructions in
commutative and non commutative algebra associated to the study
of these functions.
Beyond this into algebraic geometry
There is a very efficient approach to computations by
residue at points at infinity in the wonderful compactification
of the associated hyperplane arrangement.
This requires another talk!!!
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
We have shown that there are interesting constructions in
commutative and non commutative algebra associated to the study
of these functions.
Beyond this into algebraic geometry
There is a very efficient approach to computations by
residue at points at infinity in the wonderful compactification
of the associated hyperplane arrangement.
This requires another talk!!!
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
We have shown that there are interesting constructions in
commutative and non commutative algebra associated to the study
of these functions.
Beyond this into algebraic geometry
There is a very efficient approach to computations by
residue at points at infinity in the wonderful compactification
of the associated hyperplane arrangement.
This requires another talk!!!
Claudio Procesi.
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A particularly interesting case is when we take for A the list of
positive roots of a root system, or multiples of this list. In this case
one has applications to Clebsh–Gordan coefficients.
THE
Claudio Procesi.
END
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A particularly interesting case is when we take for A the list of
positive roots of a root system, or multiples of this list. In this case
one has applications to Clebsh–Gordan coefficients.
THE
Claudio Procesi.
END
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A particularly interesting case is when we take for A the list of
positive roots of a root system, or multiples of this list. In this case
one has applications to Clebsh–Gordan coefficients.
THE
Claudio Procesi.
END
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A particularly interesting case is when we take for A the list of
positive roots of a root system, or multiples of this list. In this case
one has applications to Clebsh–Gordan coefficients.
THE
Claudio Procesi.
END
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A particularly interesting case is when we take for A the list of
positive roots of a root system, or multiples of this list. In this case
one has applications to Clebsh–Gordan coefficients.
THE
Claudio Procesi.
END
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A particularly interesting case is when we take for A the list of
positive roots of a root system, or multiples of this list. In this case
one has applications to Clebsh–Gordan coefficients.
THE
Claudio Procesi.
END
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A particularly interesting case is when we take for A the list of
positive roots of a root system, or multiples of this list. In this case
one has applications to Clebsh–Gordan coefficients.
THE
Claudio Procesi.
END
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A particularly interesting case is when we take for A the list of
positive roots of a root system, or multiples of this list. In this case
one has applications to Clebsh–Gordan coefficients.
THE
Claudio Procesi.
END
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A particularly interesting case is when we take for A the list of
positive roots of a root system, or multiples of this list. In this case
one has applications to Clebsh–Gordan coefficients.
THE
Claudio Procesi.
END
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
A particularly interesting case is when we take for A the list of
positive roots of a root system, or multiples of this list. In this case
one has applications to Clebsh–Gordan coefficients.
THE
Claudio Procesi.
END
Topics in Hyperplane Arrangements, Polytopes and Box-Splines