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 , F p , 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 , F p , 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� ( c n ( A )) n finite dimensional over R # (multilinear) polynomials not in R

Let’s be concrete... Definition f ( x 1 , . . . , x n ) ∈ F � X � is a PI of A iff f ( a 1 , . . . , a n ) = 0 for all a i ∈ A

Let’s be concrete... Definition f ( x 1 , . . . , x n ) ∈ F � X � is a PI of A iff f ( a 1 , . . . , a n ) = 0 for all a i ∈ A � PI’s are ‘uniform relations’

Let’s be concrete... Definition f ( x 1 , . . . , x n ) ∈ F � X � is a PI of A iff f ( a 1 , . . . , a n ) = 0 for all a i ∈ A � PI’s are ‘uniform relations’ Examples: 1. A abelian iff xy − yx ≡ A 0 2. A nilpotent of degree n iff x 1 . . . x n ≡ A 0

Let’s be concrete... Definition f ( x 1 , . . . , x n ) ∈ F � X � is a PI of A iff f ( a 1 , . . . , a n ) = 0 for all a i ∈ A � PI’s are ‘uniform relations’ Examples: 1. A abelian iff xy − yx ≡ A 0 2. A nilpotent of degree n iff x 1 . . . x n ≡ A 0 3. A with dim R A = n < ∞ , then � St n +1 ( x 1 , · · · , x n +1 ) = sgn( σ ) x σ (1) . . . x σ ( n +1) σ ∈ S n +1

Guiding question How can we distinguish in terms of Id( A ) = { f ∈ F � X � | f ≡ A 0 } ?

Guiding question How can we distinguish in terms of Id( A ) = { f ∈ F � X � | f ≡ A 0 } ? Theorem There exist ( f i ) i ∈ I multilinear such that Id( A ) = ( f i | i ∈ I ) T − id .

Guiding question How can we distinguish in terms of Id( A ) = { f ∈ F � X � | f ≡ A 0 } ? Theorem There exist ( f i ) i ∈ I multilinear such that Id( A ) = ( f i | i ∈ I ) T − id . � consider P n ( F ) = span F { x σ (1) . . . x σ ( n ) | σ ∈ S n } for all n

Guiding question How can we distinguish in terms of Id( A ) = { f ∈ F � X � | f ≡ A 0 } ? Theorem There exist ( f i ) i ∈ I multilinear such that Id( A ) = ( f i | i ∈ I ) T − id . � consider P n ( F ) = span F { x σ (1) . . . x σ ( n ) | σ ∈ S n } for all n Definition P n ( F ) c n ( A ) = dim F P n ( F ) ∩ Id( A ) , the n-th codimension of A

Here (s)he is: S n ! τ ∈ S n , τ · x σ (1) . . . x σ ( n ) := x τ ( σ (1)) . . . x τ ( σ ( n ))

Here (s)he is: S n ! τ ∈ S n , τ · x σ (1) . . . x σ ( n ) := x τ ( σ (1)) . . . x τ ( σ ( n )) P n ( F ) � P n ( A ) = P n ( F ) ∩ Id( A ) is an FS n -module

Here (s)he is: S n ! τ ∈ S n , τ · x σ (1) . . . x σ ( n ) := x τ ( σ (1)) . . . x τ ( σ ( n )) P n ( F ) � P n ( A ) = P n ( F ) ∩ Id( A ) is an FS n -module In the bright char( F ) = 0 world: 1 − 1 { λ ⊢ n } − − → { simple S n -modules } �→ S ( λ ) λ and, � c n ( A ) = m λ dim F S ( λ ) λ ⊢ n

The asymptotic growth Regev’s Conjecture √ √ There exist constants t ∈ Z 2 , d ∈ N and c ∈ Q ( 2 π, b ) such that c n ( A ) ≃ cn t d n .

The asymptotic growth Regev’s Conjecture √ √ There exist constants t ∈ Z 2 , d ∈ N and c ∈ Q ( 2 π, b ) such that c n ( A ) ≃ cn t d n . 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 F p world... classicaly modular � filtrations � S ( λ ) S ( λ ) D ( λ ) = � Rad( S ( λ )) n ! dim F S ( λ ) = dim F D ( λ ) =? � � h λ ( i , j )

Some Thoughts Existence of nice Specht filtration? Does there exists a filtration P n ( R ) P n ( R ) ∩ Id( A ) ⊃ M 1 ⊃ M 2 ⊃ · · · ⊃ M l ⊃ M l +1 = { 0 } S R ( λ ) M i +1 ∼ M i with mS R ( λ ) for some λ ⊢ n and m ∈ R ? = RS n

Some Thoughts Existence of nice Specht filtration? Does there exists a filtration P n ( R ) P n ( R ) ∩ Id( A ) ⊃ M 1 ⊃ M 2 ⊃ · · · ⊃ M l ⊃ M l +1 = { 0 } S R ( λ ) M i +1 ∼ M i with mS R ( λ ) for some λ ⊢ n and m ∈ R ? = RS n Understandable ’dominant’ factors? Does there exists λ ( n ) ⊢ n such that P n ( F ) 1. D ( λ ( n ) ) factor of P n ( F ) ∩ Id( A ) for all n 2. dim F D ( λ ( n ) ) ≈ c n ( A ) for n >> 0. 3. dim F D ( λ ( n ) ) explicit