PO-set paths and q -commuting minors
Aaron Lauve
LaCIM - UQAM
Séminaire de combinatoire et d’informatique mathématique
le 7 avril 2006
http://www.lacim.uqam.ca/∼lauve (“Research”)
[email protected]
Outline
I. Flags
• what is a flag?
• who cares?
II. (commutative) Generic matrices
• column-minor identities
• homogeneous coordinate ring of the flag variety
III.
q -Generic matrices
• row-quantum-minor identities
• quantum flag variety
IV. (noncommutative) Generic matrices
• row-quasi-minor identities
• specializations
V.
q -Commuting minors
• “missing” relations
• PO-set paths
1
I.
Flags
• Fix: an integer n > 1 and a vector space V = Cn .
• Fix: a sequence of integers λ : n ≥ λ1 > λ2 > · · · > λs ≥ 0.
Definition (Flag). A flag Φ of shape λ is a chain of subspaces
Φ:
satisfying λi
0 ⊆ V 1 ( V2 ( · · · ( V s ⊆ V
= codim Vi = n − dim Vi . Denote the collection of all flags of shape λ by
F l(λ).
2
I.
Flags
• Fix: an integer n > 1 and a vector space V = Cn .
• Fix: a sequence of integers λ : n ≥ λ1 > λ2 > · · · > λs ≥ 0.
Definition (Flag). A flag Φ of shape λ is a chain of subspaces
Φ:
satisfying λi
0 ⊆ V 1 ( V2 ( · · · ( V s ⊆ V
= codim Vi = n − dim Vi . Denote the collection of all flags of shape λ by
F l(λ).
Example. Taking n
= 6 and λ = (5, 3, 2),
Φ : span {v1 } ⊆ span {v1 , v2 , v3 } ⊆ span {v1 , v2 , v3 , v4 }
is a flag when v1 , . . . , v4 are linearly independent.
3
I.
Flags
• Fix: an integer n > 1 and a vector space V = Cn .
• Fix: a sequence of integers λ : n ≥ λ1 > λ2 > · · · > λs ≥ 0.
Definition (Flag). A flag Φ of shape λ is a chain of subspaces
Φ:
satisfying λi
0 ⊆ V 1 ( V2 ( · · · ( V s ⊆ V
= codim Vi = n − dim Vi . Denote the collection of all flags of shape λ by
F l(λ).
Example. Taking n
= 6 and λ = (5, 3, 2),
Φ : span {v1 } ⊆ span {v1 , v2 , v3 } ⊆ span {v1 , v2 , v3 , v4 }
is a flag when v1 , . . . , v4 are linearly independent.
• Choose a basis B for V and express Φ as a matrix A(Φ). . .
4
I.
Flags
• Φ : span {v1 } ⊆ span {v1 , v2 , v3 } ⊆ span {v1 , v2 , v3 , v4 }

a11 a12 a13 a14 a15 a16

 a
 21 a22 a23 a24 a25 a26
A(Φ) = 
 a31 a32 a33 a34 a35 a36

a41 a42 a43 a44 a45 a46
5

V1

V
 2

 V2

V3
I.
Flags
• Φ : span {v1 } ⊆ span {v1 , v2 , v3 } ⊆ span {v1 , v2 , v3 , v4 }

a11 a12 a13 a14 a15 a16

 a
 21 a22 a23 a24 a25 a26
A(Φ) = 
 a31 a32 a33 a34 a35 a36

a41 a42 a43 a44 a45 a46
6

V1

V
 2

 V2

V3
I.
Flags
• Φ : span {v1 } ⊆ span {v1 , v2 , v3 } ⊆ span {v1 , v2 , v3 , v4 }

a11 a12 a13 a14 a15 a16

 a
 21 a22 a23 a24 a25 a26
A(Φ) = 
 a31 a32 a33 a34 a35 a36

a41 a42 a43 a44 a45 a46
7

V1

V
 2

 V2

V3
I.
Flags
• Φ : span {v1 } ⊆ span {v1 , v2 , v3 } ⊆ span {v1 , v2 , v3 , v4 }

a11 a12 a13 a14 a15 a16

 a
 21 a22 a23 a24 a25 a26
A(Φ) = 
 a31 a32 a33 a34 a35 a36

a41 a42 a43 a44 a45 a46


V1

V
 2

 V2

V3
∗

 ∗ ∗ ∗

• Unique up to change of basis! . . . multiplying by Pλ = 
 ∗ ∗ ∗

∗ ∗ ∗ ∗
8




 on the left.


I.
Who Cares?

a12 · · · · · ·
a
 11

 a21 a22
A(Φ) = 
..
 ..
 .
.

ad1 ad2 · · · · · ·
a1n
a2n
..
.
adn




 (d = n − λs )



Representation Theory Notice that GLn (C) (and its many important subgroups) permutes
F l(λ) by right-multiplication.
9
I.
Who Cares?

A(Φ) = 
a11 a12 a13 a14
a21 a22 a23 a24


Topology/Algebraic Geometry The space F l(λ) is a (projective) algebraic variety and a
CW-complex, described via Gaussian elimination (G.E.).
Consider the case n
= 4, λ = (2):
"
µ = (00)
"
#
(11)
"
#
#
G.E.
−→
Open cells:
(10)
"
#
(21)
"
#
10
(20)
"
#
(22)
"
#
I.
Who Cares?

a11 a12 a13 a14
A(Φ) = 
a21 a22 a23 a24


Topology/Algebraic Geometry The space F l(λ) is a (projective) algebraic variety and a
CW-complex, described via Gaussian elimination.
Consider the case n
= 4, λ = (2):
Schubert cells Ωµ :
µ = (00)
"
#
(11)
"
#
Partitions µ with |λ|
(10)
"
#
(21)
"
#
(20)
"
#
(22)
"
#
= 2 parts and part-size at most n − 2 = 2.
Combinatorics The classes [Ωµ ] in the cohomology ring H • (F l(λ)) are Schur
polynomials!!
11
II.
(commutative) Generic Matrices
Definition. Let X
= (xij ) be an n × n matrix of commuting indeterminants. Call X a
2
generic matrix (coordinate functions for a “generic point” in C n ).


x i1 j2 · · · x i1 jd
x

 i1 j1



x
x
·
·
·
x
i2 j1
i2 j2
i2 jd 
d
.

Definition. For I, J ∈ [n] , put XI,J = 
..
..
..


 .
.
.


x id j1 x id j2 · · · x id jd
Let XJ denote the special case I = (1, 2, . . . , d) (take the first d rows of X ).
Consider the column-minors of shape λ:
M(λ) = {[[I]] := det XI : n − |I| ∈ λ} .
Problem: Describe the relations R among the minors M(λ).
12
II.
(commutative) Generic Matrices
Answer (Schur ‘01, Hodge ‘43): The minors M(λ) satisfy the Young symmetry relations
0=
X
(−1)`(L\Λ|Λ) [[L \ Λ]][[Λ|M ]]
(YL,M )
Λ⊆L
|Λ|=r
for any 1
≤ r and any L, M ⊆ [n] with |M | + r ≤ |L| − r ∈ n − λ.
Moreover, writing M(λ)
= {I1 , . . . IN }, if F (Z1 , . . . , ZN ) is a polynomial which is
zero on substitution Zi 7→ [[Ii ]], then F is algebraically dependent on the (YL,M ).
Example:
[[12]][[34]] − [[13]][[24]] + [[23]][[14]] = 0.
Non-example: (Sylvester’s Identity)
(det A123,123 ) (det A2,2 ) = (det A12,12 ) (det A23,23 ) − (det A12,23 ) (det A23,12 ) .
13
III.
q -Generic Matrices
2
Definition. Call a matrix X q -generic if every 2 × 2 submatrix
(←) ba = q ab
dc = q cd
( ↑ ) ca = q ac
db = q bd
6
6
6
6
4
a b
c d
3
7
7
7
7
5
satisfies:
(%) cb = bc
(-) da = ad + (q − q −1 ) bc
Define the quantum determinant by
detq XI,J =
X
(−q)−`(σ) xi1 jσ(1) xi2 jσ(2) · · · xid jσ(d)
σ∈Sd
for I, J
∈ [n]d , and let [[J]] now denote detq XJ .
Problem: Describe the relations R among the minors M q (λ).
14
III.
q -Generic Matrices
Answer (Taft-Towber ‘91): The set R is (algebraically) generated by the relations below:
q -Alternating: (∀I ∈ [n]d ) [[I]] = (−q)−`(σ) [[σI]] if σ “straightens” I.
q -Young symmetry: (∀L, M ⊆ [n], r > 0) s.t. |M | + r ≤ |L| − r
X
0=
(−q)−`(L\Λ|Λ) [[L \ Λ]][[Λ | M ]] .
(YL,M )
q -Straightening: (∀I, J ( [n]) s.t. |J| < |I|
X
[[J]][[I]] =
(−q)+`(Λ|I\Λ) [[J|I \ Λ]][[Λ]] .
(SJ,I )
Λ⊂L,|Λ|=r
Λ⊆I,|Λ|=|J|
Example:
[[12]][[34]] − 1q [[13]][[24]] +
Non-examples: (cf. Goodearl, ‘05)
15
1
[[23]][[14]]
q2
= 0.
IV.
(noncommutative) Generic Matrices
• Now let X be a matrix of noncommuting variables.
Definition (Gelfand-Retakh ‘91). The (ij)-quasideterminant |X| ij is defined whenever X ij
is invertible, and in that case,
|X|ij
= 16
IV.
(noncommutative) Generic Matrices
• Now let X be a matrix of noncommuting variables.
Definition (Gelfand-Retakh ‘91). The (ij)-quasideterminant |X| ij is defined whenever X ij
is invertible, and in that case,
|X|ij
= =
−
·
• 2 × 2 Example:
|A|12
a11 a12
= a21 a22
= a12 − a11 a−1 a22 .
21
17
−1
·
IV.
(noncommutative) Generic Matrices
• It is better to take ratios of quasideterminants as “column-minor” replacements
Definition. Given an n × n matrix X and a partition λ, the column-minors are given by
n
−1
Mquasi (λ) = pK
ij (X) := |Xi∪K |di |Xj∪K |dj
o
i, j ∈ [n], K ⊆ [n] \ i, n − 1 − |K| ∈ λ
Quasi-Plücker Relations (Gelfand-Retakh ‘97, L. ’04): If L, M
⊆ [n], i ∈ [n] \ M ,
|M | ≤ |L| − 1, then:
1=
X
L\j
pM
ij (X)pji (X) .
j∈L
Problem: Describe the relations R satisfied by Mquasi
• Need to expand search to rational expressions, not just algebraic ones.
• some are known. . . maybe all?
18
(Pi,L,M )
IV.
(noncommutative) Generic Matrices
Why study these matrices?
• May study “all” the deformations of the commutative generic matrix at once!
• X → commutative generic:
• X → q -generic:
−1
pK
ij → ratio of two determinants [[i ∪ K]] [[j ∪ K]]
−1
pK
ij → ratio of two quantum determinants [[i ∪ K]] [[j ∪ K]]
Question: How much of R(M) or R(Mq ) may be recovered on specializing X and
R(Mquasi )?
Answer: For commutative case, all. For q -generic case, a lot. . .
19
V.
“Missing” Relations
• While showing R(Mquasi ) ⇒ (YL,M ) for quantum setting, we discover something “new:”
• [[I]] and [[J]] often q -commute [Krob-Leclerc ‘95].
Definition. Given two subsets I, J
⊆ [n], we say J surrounds I , written J y I , if (i)
|J| ≤ |I|, and (ii) there exist disjoint subsets ∅ ⊆ J 0 , J 00 ⊆ J such that:
a.
J \ I = J 0 ∪˙ J 00 ,
b.
j 0 < i for all j 0 ∈ J 0 and i ∈ I \ J ,
c.
i < j 00 for all i ∈ I \ J and j 00 ∈ J 0 ,
Theorem (q -Commuting Minors). If the subsets I, J
⊆ [n] satisfy J y I , the quantum
minors [[J]] and [[I]] q -commute. Specifically,
[[J]][[I]] = q |J
00 |−|J 0 |
20
[[I]][[J]] .
(CJ,I )
V.
Finding the Missing Relations
Theorem (L, ‘06). Given I, J
⊆ [n]. If J y I , then the sum CJ,I may be written as a
Z[q, q −1 ]-linear combination of the sums SJ,I and YI∪(J\K),K ( ∅ ⊆ K ( J ). The
coefficients involved are described in terms of path-weighs in a directed graph.
Open Question (converse): Give an easy proof that only if J
[Leclerc-Zelevinsky ‘98]
21
y I can C J,I be so written.
V.
Example 1
Writing C1,234 in terms of S1,234 and YL,M .
−q −1 f234 f1
C1,234
f1 f234
S1,234
f1 f234 −q 2 f123 f4
q 2 Y1234,∅
+q 1 f124 f3
−q 0 f134 f2
f123 f4 −q −1 f124 f3 +q −2 f134 f2 −q −3 f234 f1
22
V.
Example 1
Writing C1,234 in terms of S1,234 and YL,M .
−q −1 f234 f1
C1,234
f1 f234
S1,234
f1 f234 −q 2 f123 f4
q 2 Y1234,∅
+q 1 f124 f3
−q 0 f134 f2
q 2 f123 f4 −q 1 f124 f3 +q 0 f134 f2 −q −1 f234 f1
C1,234 = S1,234 + q 2 Y1234,∅
23
V.
Example 2
Writing C14,23 in terms of YL,M .
[14][23] [24][13] [34][12] [12][34] [13][24] [23][14]
−1
C14,23
1
Y1234,∅
q −2
−q −3
q −4
Y234,1
0
q −2
−q −3
Y123,4
0
1
q −2
−q −1
1
24
−q −1
−q −1
q −2
V.
Example 2
Writing C14,23 in terms of YL,M .
[14][23] [24][13] [34][12] [12][34] [13][24] [23][14]
−1
C14,23
1
Y1234,∅
q −2
−q −3
q −4
Y234,1
0
q −2
−q −3
Y123,4
0
1
q −2
−q −1
1
25
−q −1
−q −1
q −2
V.
Example 2
Writing C14,23 in terms of YL,M .
[14][23] [24][13] [34][12] [12][34] [13][24] [23][14]
C14,23
1
Y1234,∅
q −2
Y234,1
−1
−q −3
q −4
q −2
−q −3
1
26
q −2
−q −1
1
Y123,4
−q −1
−q −1
q −2
V.
Example 2
Writing C14,23 in terms of YL,M .
[14][23] [24][13] [34][12] [12][34] [13][24] [23][14]
C14,23
1
Y1234,∅
q −2
Y234,1
−1
−q −3
q −4
q −2
−q −3
1
27
q −2
−q −1
1
Y123,4
−q −1
−q −1
q −2
V.
Example 2
Writing C14,23 in terms of YL,M .
[14][23] [24][13] [34][12] [12][34] [13][24] [23][14]
C14,23
1
Y1234,∅
q −2
Y234,1
−1
−q −3
q −4
q −2
−q −3
1
28
q −2
−q −1
1
Y123,4
−q −1
−q −1
q −2
V.
Example 3
Writing C156,234 in terms of YL,M .
y∅
y1
y5
y6
e∅
e1
1
α∅1 α∅5 α∅6 α∅15 α∅16 α∅56 α∅156
e5
e6
e15
e16
e56
α115 α116
1
α515
1
e156
α1156
α556 α5156
α616 α656 α6156
1
y 15
156
α15
1
y 16
156
α16
1
y 56
1
29
156
α56
V.
Example 3
Writing C156,234 in terms of YL,M .
y∅
y1
e∅
e1
1
α∅1 α∅5 α∅6 α∅15 α∅16 α∅56 α∅156
e5
e6
e15
y6
e56
α115 α116
1
y5
e16
α515
1
e156
α1156
α556 α5156
α616 α656 α6156
1
y 15
156
α15
1
y 16
156
α16
1
y 56
1
• Perform Gaussian Elimination!!
• Hope that what remains on the first row is 1 and (−q)θ q 2−1
30
156
α56
V.
Example 3
Writing C156,234 in terms of YL,M .
y∅
y1
y5
y6
e∅
e1
1
α∅1 α∅5 α∅6 α∅15 α∅16 α∅56 α∅156
e5
e6
e15
e16
e56
α115 α116
1
α515
1
e156
α1156
α556 α5156
α616 α656 α6156
1
y 15
156
α15
1
y 16
156
α16
1
y 56
1
• Perform Gaussian Elimination!!
X
J
J
J
• In general, we get: coeff e = α∅ −
α∅A αA
+
∅(A(J
31
156
α56
X
∅(A(B(J
B J
α∅A αA
αB − · · ·
V.
Example 3
A directed graph with edge-weights Γ(J; I) associated to J
56
6
156
16
5
∅
α1∅
= 156 and I = 234.
α156
15
−→
15
1
α15
1
32
156 = −q −5
α∅1 α115 α15
V.
Example 3
A directed graph with edge-weights Γ(J; I) associated to J
56
6
156
16
5
∅
α1∅
= 156 and I = 234.
α156
15
−→
15
1
α15
1
33
156 = −q −5
α∅1 α115 α15
V.
Example 3
A directed graph with edge-weights Γ(J; I) associated to J
56
6
156
16
5
∅
B
αA
B C
αA
αB
1
:= (−q)
=
α156
15
−→
15
α1∅
h
= 156 and I = 234.
156 = −q −5
α∅1 α115 α15
α15
1
−`(J\B|B\A)−`(B\A|A)+{2|J\B|−|I|}·|(B\A)∩J 0 |
(−q)
2`((B\A)∩J 0 |C\B)−2`(C\B|(B\A)∩J 00 )
34
i
C
αA
V.
Let J
PO-set paths
= J 0 ∪ J 00 = {j10 , . . . , jr0 0 } ∪ {j100 , . . . , jr0000 }. We consider certain paths in Γ(J; I):
n
o
P0 = (A1 , A2 , . . . , Ap ) | Ai ⊆ J s.t. ∅ ( A1 ( A2 ( · · · ( Ap ( J
and P
= P0 ∪ 0̂ ∪ 1̂, where 0̂ = (∅), and
1̂ = ({j100 }, {j100 , j200 }, . . . , J 00 , {jr0 0 , . . . , jr0000 }, . . . , {j20 , . . . , jr0000 }, J).
The weight
α(π) of a path π = (A1 , . . . , Ap ) ∈ P is the product of edge weights of the
augmented path (∅, π, J):
A
p
A2
J
α∅A1 · αA
·
·
·
α
·
α
.
A
A
p
1
p−1

 min(K ∩ J 0 ) if K ∩ J 0 6= ∅
Definition. For K ⊆ J , define mM(K) :=
 max(K ∩ J 00 ) otherwise.
For any path π
= (A1 , . . . , Ap ), put A0 = ∅ and Ap+1 = J .
35
V.
PO-set paths
Definition. Call 1̂ regular and 0̂ irregular. Call a path (A1 , . . . , Ap )
∈ P0 regular (or regular
at position i0 ), if (∃i0 ) (1
≤ i0 ≤ p) satisfying:
(b) Ai0 \ Ai0 −1 = mM(Ai0 +1 \ Ai0 −1 ).
(a) |Ai | = i (∀ 1 ≤ i ≤ i0 );
A path in P0 is irregular if it is nowhere regular.
Proposition. The regular and irregular subsets of P are equinumerous.
A bijection ℘. Given an irregular path π
= (A1 , . . . , Ap ) ∈ P0 , we insert a new set B so
that ℘(π) is regular at B :
1. Find the unique i0 satisfying:
2. Compute b
3. Put B
(|Ai | = i ∀i ≤ i0 ) ∧ (|Ai0 +1 | > i0 + 1).
= mM(Ai0 +1 \ Ai0 )
= Ai0 ∪ {b}.
4. Define ℘(π)
Finally, put ℘(0̂)
:= (A1 , . . . , Ai0 , B, Ai0 +1 , . . . , Ap ).
= ({j1 }).
Proposition. The bijection ℘ is path-weight preserving and α(℘ −1 (1̂))
36
= (−q)θ q |J
00 |−|J 0 |
.
56
156
16
6
5
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