Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
ZSn -modules and polynomial identities with
integer coefficients
Alexey Gordienko,
Geoffrey Janssens
July 29, 2014
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
1 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Outline
1
Introduction
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
2
Relations between different codimensions of algebras
3
Structure as ZSn -modules
ZSn -modules
Analog of Drensky’s theorem
A particular case of the Littlewood — Richardson rule
4
Examples
5
Further research
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
2 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Polynomial identities with coefficients from a field
Let F be a field of charF =0 and A an algebra.
definition
A polynomial f = f (x1 , . . . , xn ) ∈ F hX i is a polynomial identity
(PI) of A iff f (a1 , . . . , an ) = 0 for all a1 , . . . , an ∈ A.
Examples:
[x, y ] = xy − yx ≡ 0 is a PI of A iff A is a commutative
algebra.
x m ≡ 0 is PI of A iff A is nil of bounded index.
x1 . . . . .xm ≡ 0 iff A is nilpotent.
All matrix algebras Mn (F ) satisfy the ’standard identity’.
Also all f.d algebras are PI
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
3 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Polynomial identities with coefficients from a field
Let F be a field of charF =0 and A an algebra.
definition
A polynomial f = f (x1 , . . . , xn ) ∈ F hX i is a polynomial identity
(PI) of A iff f (a1 , . . . , an ) = 0 for all a1 , . . . , an ∈ A.
Examples:
[x, y ] = xy − yx ≡ 0 is a PI of A iff A is a commutative
algebra.
x m ≡ 0 is PI of A iff A is nil of bounded index.
x1 . . . . .xm ≡ 0 iff A is nilpotent.
All matrix algebras Mn (F ) satisfy the ’standard identity’.
Also all f.d algebras are PI
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
3 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Polynomial identities with coefficients from a field
Let F be a field of charF =0 and A an algebra.
definition
A polynomial f = f (x1 , . . . , xn ) ∈ F hX i is a polynomial identity
(PI) of A iff f (a1 , . . . , an ) = 0 for all a1 , . . . , an ∈ A.
Examples:
[x, y ] = xy − yx ≡ 0 is a PI of A iff A is a commutative
algebra.
x m ≡ 0 is PI of A iff A is nil of bounded index.
x1 . . . . .xm ≡ 0 iff A is nilpotent.
All matrix algebras Mn (F ) satisfy the ’standard identity’.
Also all f.d algebras are PI
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
3 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Polynomial identities with coefficients from a field
Let F be a field of charF =0 and A an algebra.
definition
A polynomial f = f (x1 , . . . , xn ) ∈ F hX i is a polynomial identity
(PI) of A iff f (a1 , . . . , an ) = 0 for all a1 , . . . , an ∈ A.
Examples:
[x, y ] = xy − yx ≡ 0 is a PI of A iff A is a commutative
algebra.
x m ≡ 0 is PI of A iff A is nil of bounded index.
x1 . . . . .xm ≡ 0 iff A is nilpotent.
All matrix algebras Mn (F ) satisfy the ’standard identity’.
Also all f.d algebras are PI
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
3 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Polynomial identities with coefficients from a field
Let F be a field of charF =0 and A an algebra.
definition
A polynomial f = f (x1 , . . . , xn ) ∈ F hX i is a polynomial identity
(PI) of A iff f (a1 , . . . , an ) = 0 for all a1 , . . . , an ∈ A.
Examples:
[x, y ] = xy − yx ≡ 0 is a PI of A iff A is a commutative
algebra.
x m ≡ 0 is PI of A iff A is nil of bounded index.
x1 . . . . .xm ≡ 0 iff A is nilpotent.
All matrix algebras Mn (F ) satisfy the ’standard identity’.
Also all f.d algebras are PI
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
3 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
PI’s follow from multilinear ones
Thus, the PI’s are strongly connected with the algebraic
structure of the algebra A. Denote, Id(A, F ) the set of all PI’s of
A over F .
definition
The space P
of multilinear polynomials:
aσ xσ(1) . . . xσ(n) | aσ ∈ F }.
Pn (F ) = {
σ∈Sn
proposition
If char F = 0 then all PI’s, Id(A, F ), are consequences of
multilinear PI’s ( Id(A, F ) ∩ Pn (F )).
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
4 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
PI’s follow from multilinear ones
Thus, the PI’s are strongly connected with the algebraic
structure of the algebra A. Denote, Id(A, F ) the set of all PI’s of
A over F .
definition
The space P
of multilinear polynomials:
aσ xσ(1) . . . xσ(n) | aσ ∈ F }.
Pn (F ) = {
σ∈Sn
proposition
If char F = 0 then all PI’s, Id(A, F ), are consequences of
multilinear PI’s ( Id(A, F ) ∩ Pn (F )).
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
4 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
PI’s follow from multilinear ones
Thus, the PI’s are strongly connected with the algebraic
structure of the algebra A. Denote, Id(A, F ) the set of all PI’s of
A over F .
definition
The space P
of multilinear polynomials:
aσ xσ(1) . . . xσ(n) | aσ ∈ F }.
Pn (F ) = {
σ∈Sn
proposition
If char F = 0 then all PI’s, Id(A, F ), are consequences of
multilinear PI’s ( Id(A, F ) ∩ Pn (F )).
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
4 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
PI’s follow from multilinear ones
Thus, the PI’s are strongly connected with the algebraic
structure of the algebra A. Denote, Id(A, F ) the set of all PI’s of
A over F .
definition
The space P
of multilinear polynomials:
aσ xσ(1) . . . xσ(n) | aσ ∈ F }.
Pn (F ) = {
σ∈Sn
proposition
If char F = 0 then all PI’s, Id(A, F ), are consequences of
multilinear PI’s ( Id(A, F ) ∩ Pn (F )).
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
4 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Codimensions over a field
More interesting to look at is the F -module:
Pn (F )
.
Pn (F ) ∩ Id(A, F )
definition
The non-negative integer cn (A, F ) = dimF
the nth codimension of the algebra A.
Pn (F )
Pn (F )∩Id(A,F )
is called
Remark:
If ∃n ∈ N s.t cn (A, F ) < n! then A is PI.
In this spirit, Regev: if A1 and A2 are PI, then A1 ⊗ A2 is PI.
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
5 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Codimensions over a field
More interesting to look at is the F -module:
Pn (F )
.
Pn (F ) ∩ Id(A, F )
definition
The non-negative integer cn (A, F ) = dimF
the nth codimension of the algebra A.
Pn (F )
Pn (F )∩Id(A,F )
is called
Remark:
If ∃n ∈ N s.t cn (A, F ) < n! then A is PI.
In this spirit, Regev: if A1 and A2 are PI, then A1 ⊗ A2 is PI.
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
5 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Codimensions over a field
More interesting to look at is the F -module:
Pn (F )
.
Pn (F ) ∩ Id(A, F )
definition
The non-negative integer cn (A, F ) = dimF
the nth codimension of the algebra A.
Pn (F )
Pn (F )∩Id(A,F )
is called
Remark:
If ∃n ∈ N s.t cn (A, F ) < n! then A is PI.
In this spirit, Regev: if A1 and A2 are PI, then A1 ⊗ A2 is PI.
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
5 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Conjectures
Conjecture (S.A. Amitsur)
If A is a PI-algebra over p
a field of characteristic 0, then there
exists PIexp(A) := lim n cn (A, F ) ∈ Z+ .
n→∞
Example (A. Regev)
p
lim n cn (Mk (F ), F ) = k 2 where Mk (F ) is the algebra of k × k
n→∞
matrices.
S.A. Amitsur’s conjecture was proved
in 1999 by A. Giambruno and M.V. Zaicev for
codimensions of associative algebras;
goal: develop a similar theory for rings.
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
6 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Conjectures
Conjecture (S.A. Amitsur)
If A is a PI-algebra over p
a field of characteristic 0, then there
exists PIexp(A) := lim n cn (A, F ) ∈ Z+ .
n→∞
Example (A. Regev)
p
lim n cn (Mk (F ), F ) = k 2 where Mk (F ) is the algebra of k × k
n→∞
matrices.
S.A. Amitsur’s conjecture was proved
in 1999 by A. Giambruno and M.V. Zaicev for
codimensions of associative algebras;
goal: develop a similar theory for rings.
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
6 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Conjectures
Conjecture (S.A. Amitsur)
If A is a PI-algebra over p
a field of characteristic 0, then there
exists PIexp(A) := lim n cn (A, F ) ∈ Z+ .
n→∞
Example (A. Regev)
p
lim n cn (Mk (F ), F ) = k 2 where Mk (F ) is the algebra of k × k
n→∞
matrices.
S.A. Amitsur’s conjecture was proved
in 1999 by A. Giambruno and M.V. Zaicev for
codimensions of associative algebras;
goal: develop a similar theory for rings.
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
6 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Conjectures
Conjecture (S.A. Amitsur)
If A is a PI-algebra over p
a field of characteristic 0, then there
exists PIexp(A) := lim n cn (A, F ) ∈ Z+ .
n→∞
Example (A. Regev)
p
lim n cn (Mk (F ), F ) = k 2 where Mk (F ) is the algebra of k × k
n→∞
matrices.
S.A. Amitsur’s conjecture was proved
in 1999 by A. Giambruno and M.V. Zaicev for
codimensions of associative algebras;
goal: develop a similar theory for rings.
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
6 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Polynomial identities with integer coefficients
Let R be a ring. Basic definitions with Z instead of F can be
taken over:
f ∈ ZhX i is a polynomial identity of a ring R with integer
coefficients if f (a1 , . . . , an ) = 0 for all ai ∈ R.
Denote the set of polynomial identities of R with integer
coefficients by Id(R, Z).
Denote the set ofP
multiliniear polynomials over Z by Pn (Z).
Thus, Pn (Z) = {
aσ xσ(1) . . . xσ(n) | aσ ∈ Z}.
σ∈Sn
The quotient:
Pn (Z)/Pn (Z) ∩ Id(R, Z)
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
7 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Polynomial identities with coefficients from a field
Polynomial identities with integer coefficients
Z-codimensions
Pn (Z)
is a finitely generated Abelian group
Then Pn (Z)∩Id(R,Z)
which is a direct sum of free and primary cyclic groups:
Pn (Z)
∼
. . ⊕ Z} ⊕
= Z ⊕ .{z
Pn (Z) ∩ Id(R, Z) |
cn (R,0)
M
M
Zpk ⊕ . . . ⊕ Zpk .
|
{z
}
p is a prime k ∈N
number
cn (R,pk )
We call the numbers cn (R, q) the codimensions of
polynomial identities of R with integer coefficients.
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
8 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Z-codimensions of algebras
Two natural questions arise:
1
If R = A an algebra over F what is the connection between
cn (A, F ) and cn (A, pk )?
2
If R is a ring, what is the connection between
cn (R ⊗Z F , F ) and cn (R, pk )?
Proposition (Question 1)
Let A be an algebra over a field F . Then cn (A, q) = 0 for all
n ∈ N and q 6= char F . Moreover cn (A, F ) 6 cn (A, char F ) for all
n ∈ N and if F = Q or Zp then equality occurs .
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
9 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Z-codimensions of algebras
Two natural questions arise:
1
If R = A an algebra over F what is the connection between
cn (A, F ) and cn (A, pk )?
2
If R is a ring, what is the connection between
cn (R ⊗Z F , F ) and cn (R, pk )?
Proposition (Question 1)
Let A be an algebra over a field F . Then cn (A, q) = 0 for all
n ∈ N and q 6= char F . Moreover cn (A, F ) 6 cn (A, char F ) for all
n ∈ N and if F = Q or Zp then equality occurs .
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
9 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Z-codimensions of algebras
Two natural questions arise:
1
If R = A an algebra over F what is the connection between
cn (A, F ) and cn (A, pk )?
2
If R is a ring, what is the connection between
cn (R ⊗Z F , F ) and cn (R, pk )?
Proposition (Question 1)
Let A be an algebra over a field F . Then cn (A, q) = 0 for all
n ∈ N and q 6= char F . Moreover cn (A, F ) 6 cn (A, char F ) for all
n ∈ N and if F = Q or Zp then equality occurs .
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
9 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Z-codimensions of algebras
Two natural questions arise:
1
If R = A an algebra over F what is the connection between
cn (A, F ) and cn (A, pk )?
2
If R is a ring, what is the connection between
cn (R ⊗Z F , F ) and cn (R, pk )?
Proposition (Question 1)
Let A be an algebra over a field F . Then cn (A, q) = 0 for all
n ∈ N and q 6= char F . Moreover cn (A, F ) 6 cn (A, char F ) for all
n ∈ N and if F = Q or Zp then equality occurs .
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
9 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Extension of a ring to an algebra over a field
The next result is concerned with the extension of a ring to an
algebra over a field (question 2):
Theorem
Let R be a ring and let F be a field. Then
cn (R/Tor R, 0) if
cn (R ⊗Z F , F ) =
cn (R/pR, p) if
char F = 0,
char F = p
where Tor R := {r ∈ R | mr = 0 for some m ∈ N} is the torsion
of R.
As corollary one can get/prove :
Amitsur’s conjecture is true for torsion-free PI rings.
Asymptotices for cn (Mn (Z), 0).
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
10 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Extension of a ring to an algebra over a field
The next result is concerned with the extension of a ring to an
algebra over a field (question 2):
Theorem
Let R be a ring and let F be a field. Then
cn (R/Tor R, 0) if
cn (R ⊗Z F , F ) =
cn (R/pR, p) if
char F = 0,
char F = p
where Tor R := {r ∈ R | mr = 0 for some m ∈ N} is the torsion
of R.
As corollary one can get/prove :
Amitsur’s conjecture is true for torsion-free PI rings.
Asymptotices for cn (Mn (Z), 0).
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
10 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
ZSn -modules
Analog of Drensky’s theorem
A particular case of the Littlewood — Richardson rule
ZSn -action
Note that the symmetric group Sn is acting on
by permutations of variables, i.e.
ZSn -module.
Pn (Z)
Pn (Z)∩Id(R,Z)
Pn (Z)
Pn (Z)∩Id(R,Z)
is an
Remember, over a field F with char F = 0, there are 1 − 1
correspondences:
Irr(Sn ) ←→ partitions λ of n ←→ Specht Modules S(λ)
Problem over Z: the Specht modules S(λ) are no longer
irreducible and even contain no irreducible ZSn -submodules.
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
11 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
ZSn -modules
Analog of Drensky’s theorem
A particular case of the Littlewood — Richardson rule
ZSn -action
Note that the symmetric group Sn is acting on
by permutations of variables, i.e.
ZSn -module.
Pn (Z)
Pn (Z)∩Id(R,Z)
Pn (Z)
Pn (Z)∩Id(R,Z)
is an
Remember, over a field F with char F = 0, there are 1 − 1
correspondences:
Irr(Sn ) ←→ partitions λ of n ←→ Specht Modules S(λ)
Problem over Z: the Specht modules S(λ) are no longer
irreducible and even contain no irreducible ZSn -submodules.
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
11 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
ZSn -modules
Analog of Drensky’s theorem
A particular case of the Littlewood — Richardson rule
ZSn -action
Note that the symmetric group Sn is acting on
by permutations of variables, i.e.
ZSn -module.
Pn (Z)
Pn (Z)∩Id(R,Z)
Pn (Z)
Pn (Z)∩Id(R,Z)
is an
Remember, over a field F with char F = 0, there are 1 − 1
correspondences:
Irr(Sn ) ←→ partitions λ of n ←→ Specht Modules S(λ)
Problem over Z: the Specht modules S(λ) are no longer
irreducible and even contain no irreducible ZSn -submodules.
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
11 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
ZSn -modules
Analog of Drensky’s theorem
A particular case of the Littlewood — Richardson rule
Analog of Drensky’s theorem
Let R be a unitary ring.
Consider for every n ∈ N the series of ZSn -submodules
Pn (Z)
% M2 ⊇ M3 ⊇ . . . ⊇ Mn ∼
=
Pn (Z) ∩ Id(R, Z)
Γn (Z)
∼
=
Γn (Z) ∩ Id(R, Z)
L
where each Mk is the image of nt=k ZSn (x1 . . . xn−t Γt (Z))
and Mn+1 := 0.
M0 :=
In other words, Mk is a span of images of
xi1 . . . xin−t [. . . , . . .] . . . [. . . , . . .] for t > k .
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
12 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
ZSn -modules
Analog of Drensky’s theorem
A particular case of the Littlewood — Richardson rule
Analog of Drensky’s theorem
Let R be a unitary ring.
Consider for every n ∈ N the series of ZSn -submodules
Pn (Z)
% M2 ⊇ M3 ⊇ . . . ⊇ Mn ∼
=
Pn (Z) ∩ Id(R, Z)
Γn (Z)
∼
=
Γn (Z) ∩ Id(R, Z)
L
where each Mk is the image of nt=k ZSn (x1 . . . xn−t Γt (Z))
and Mn+1 := 0.
M0 :=
In other words, Mk is a span of images of
xi1 . . . xin−t [. . . , . . .] . . . [. . . , . . .] for t > k .
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
12 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
ZSn -modules
Analog of Drensky’s theorem
A particular case of the Littlewood — Richardson rule
Analog of Drensky’s theorem
Then M0 /M2 ∼
= Z` (trivial Sn -action) where
` := min{n ∈ N | na = 0 for all a ∈ R}
,
Γt (Z)
⊗Z Z ↑ Sn :=
Γt (Z) ∩ Id(R, Z)
Γt (Z)
⊗Z Z
ZSn ⊗Z(St ×Sn−t )
Γt (Z) ∩ Id(R, Z)
Mt /Mt+1 ∼
=
for all 2 6 t 6 n where Sn−t is permuting xt+1 , . . . , xn and Z is a
trivial ZSn−t -module.
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
13 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
ZSn -modules
Analog of Drensky’s theorem
A particular case of the Littlewood — Richardson rule
A particular case of the Littlewood — Richardson rule
We prove the Z-analog of what sometimes is referred to as
Young’s rule and sometimes as Pieri’s formula.
Theorem
Let t, n ∈ N, m ∈ Z+ , t < n, and λ ` t and Z be the trivial
ZSn−t -module. Then
S(λ)/mS(λ) ↑ Sn := ZSn ⊗Z(St ×Sn−t ) ( S(λ)/mS(λ) ⊗Z Z)
has a series of submodules with factors S(ν)/mS(ν) where ν
runs over the set of all partitions ν ` n such that
λn 6 νn 6 λn−1 6 νn−1 6 . . . 6 λ2 6 ν2 6 λ1 6 ν1 .
(Each factor occurs exactly once.)
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
14 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
ZSn -modules
Analog of Drensky’s theorem
A particular case of the Littlewood — Richardson rule
A particular case of the Littlewood — Richardson rule
We prove the Z-analog of what sometimes is referred to as
Young’s rule and sometimes as Pieri’s formula.
Theorem
Let t, n ∈ N, m ∈ Z+ , t < n, and λ ` t and Z be the trivial
ZSn−t -module. Then
S(λ)/mS(λ) ↑ Sn := ZSn ⊗Z(St ×Sn−t ) ( S(λ)/mS(λ) ⊗Z Z)
has a series of submodules with factors S(ν)/mS(ν) where ν
runs over the set of all partitions ν ` n such that
λn 6 νn 6 λn−1 6 νn−1 6 . . . 6 λ2 6 ν2 6 λ1 6 ν1 .
(Each factor occurs exactly once.)
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
14 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Grassman algebra and upper triangular matrices
Let R1 , R2 , R be commutative rings with 1 and M a
(R1 , R2 )-bimodule. Following two examples are determined.
The generalized upper triangular matrices:
R1 M
R=
.
0 R2
The Grassman algebra
GR = he1 , e2 , . . . | ei ej = −ej ei , i, j ∈ NiR−alg
In the both cases the ZSn -module structure is described by a
’Specht Series’.
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
15 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Further research and reference
Study Amitsur’s conjecture for classes of finite rings, e.g
Mn (R) for R a finite ring.
Pn (Z)
For which classes of rings do Pn (Z)∩Id(R,Z)
can be
described by such a ’Specht Series’?
Reference of the article:
A. Gordienko, G. Janssens, ZSn -modules and polynomial
identities with integer coefficients, Int. J. Algebra Comput. 23
(2013),no. 8, 1925-1943
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
16 / 16
Introduction
Relations between different codimensions of algebras
Structure as ZSn -modules
Examples
Further research
Further research and reference
Study Amitsur’s conjecture for classes of finite rings, e.g
Mn (R) for R a finite ring.
Pn (Z)
For which classes of rings do Pn (Z)∩Id(R,Z)
can be
described by such a ’Specht Series’?
Reference of the article:
A. Gordienko, G. Janssens, ZSn -modules and polynomial
identities with integer coefficients, Int. J. Algebra Comput. 23
(2013),no. 8, 1925-1943
Alexey Gordienko, Geoffrey Janssens
ZSn -modules
16 / 16