How can (modular) representation theorists help ring theory? - - PowerPoint PPT Presentation

How can (modular) representation theorists help ring theory?

Geoffrey Janssens Free University of Brussels

R a principal ideal domain (e.g. Z, Fp, C) R-algebras A and B

Distinguishing Problem

Find invariants that detect whether A and B are isomorphic.

R a principal ideal domain (e.g. Z, Fp, C) R-algebras A and B

Distinguishing Problem

Find invariants that detect whether A and B are isomorphic. Asymptotic Ring Theory’s philosophy

A = S | R finite dimensional over R (cn(A))n # (multilinear) polynomials not in R

Let’s be concrete...

Definition

f (x1, . . . , xn) ∈ FX is a PI of A iff f (a1, . . . , an) = 0 for all ai ∈ A

Let’s be concrete...

Definition

f (x1, . . . , xn) ∈ FX is a PI of A iff f (a1, . . . , an) = 0 for all ai ∈ A PI’s are ‘uniform relations’

Let’s be concrete...

Definition

f (x1, . . . , xn) ∈ FX is a PI of A iff f (a1, . . . , an) = 0 for all ai ∈ A PI’s are ‘uniform relations’ Examples:

• 1. A abelian iff xy − yx ≡A 0
• 2. A nilpotent of degree n iff x1 . . . xn ≡A 0

Let’s be concrete...

Definition

f (x1, . . . , xn) ∈ FX is a PI of A iff f (a1, . . . , an) = 0 for all ai ∈ A PI’s are ‘uniform relations’ Examples:

• 1. A abelian iff xy − yx ≡A 0
• 2. A nilpotent of degree n iff x1 . . . xn ≡A 0
• 3. A with dimR A = n < ∞, then

Stn+1(x1, · · · , xn+1) =

• σ∈Sn+1

sgn(σ)xσ(1) . . . xσ(n+1)

Guiding question

How can we distinguish in terms of Id(A) = {f ∈ FX | f ≡A 0}?

Guiding question

How can we distinguish in terms of Id(A) = {f ∈ FX | f ≡A 0}?

Theorem

There exist (fi)i∈I multilinear such that Id(A) = (fi | i ∈ I)T−id.

Guiding question

How can we distinguish in terms of Id(A) = {f ∈ FX | f ≡A 0}?

Theorem

There exist (fi)i∈I multilinear such that Id(A) = (fi | i ∈ I)T−id. consider Pn(F) = spanF{xσ(1) . . . xσ(n) | σ ∈ Sn} for all n

Guiding question

How can we distinguish in terms of Id(A) = {f ∈ FX | f ≡A 0}?

Theorem

There exist (fi)i∈I multilinear such that Id(A) = (fi | i ∈ I)T−id. consider Pn(F) = spanF{xσ(1) . . . xσ(n) | σ ∈ Sn} for all n

Definition

cn(A) = dimF

Pn(F) Pn(F)∩Id(A), the n-th codimension of A

Here (s)he is: Sn!

τ ∈ Sn, τ · xσ(1) . . . xσ(n) := xτ(σ(1)) . . . xτ(σ(n))

Here (s)he is: Sn!

τ ∈ Sn, τ · xσ(1) . . . xσ(n) := xτ(σ(1)) . . . xτ(σ(n)) Pn(A) =

Pn(F) Pn(F)∩Id(A) is an FSn-module

Here (s)he is: Sn!

τ ∈ Sn, τ · xσ(1) . . . xσ(n) := xτ(σ(1)) . . . xτ(σ(n)) Pn(A) =

Pn(F) Pn(F)∩Id(A) is an FSn-module

In the bright char(F) = 0 world:

{λ ⊢ n}

1−1

− − → { simple Sn-modules } λ → S(λ) and, cn(A) =

• λ⊢n

mλ dimF S(λ)

The asymptotic growth

Regev’s Conjecture

There exist constants t ∈ Z

2 , d ∈ N and c ∈ Q(

√ 2π, √ b) such that cn(A) ≃ cntdn.

The asymptotic growth

Regev’s Conjecture

There exist constants t ∈ Z

2 , d ∈ N and c ∈ Q(

√ 2π, √ b) such that cn(A) ≃ cntdn. Some milestones:

• 1. Giambruno-Zaicev (1999): existence and integrality of d + explicit

algebraic formula

• 2. Berele-Regev (2008): proof for unital A but no explicit formula
• 3. Aljadeff-J.-Karasik (2017): an explicit algebraic formula for t

In the foggy Fp world...

classicaly modular

• filtrations

S(λ)

• D(λ) =

S(λ) Rad(S(λ))

dimF S(λ) =

n!

• hλ(i,j)
• dimF D(λ) =?

Some Thoughts

Existence of nice Specht filtration?

Does there exists a filtration Pn(R) Pn(R) ∩ Id(A) ⊃ M1 ⊃ M2 ⊃ · · · ⊃ Ml ⊃ Ml+1 = {0} with

Mi Mi+1 ∼

=RSn

SR(λ) mSR(λ) for some λ ⊢ n and m ∈ R?

Some Thoughts

Existence of nice Specht filtration?

Does there exists a filtration Pn(R) Pn(R) ∩ Id(A) ⊃ M1 ⊃ M2 ⊃ · · · ⊃ Ml ⊃ Ml+1 = {0} with

Mi Mi+1 ∼

=RSn

SR(λ) mSR(λ) for some λ ⊢ n and m ∈ R?

Understandable ’dominant’ factors?

Does there exists λ(n) ⊢ n such that

• 1. D(λ(n)) factor of

Pn(F) Pn(F)∩Id(A) for all n

• 2. dimF D(λ(n)) ≈ cn(A) for n >> 0.
• 3. dimF D(λ(n)) explicit