Characterization of queer super crystals Anne Schilling Department - PowerPoint PPT Presentation

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Characterization of queer super crystals Anne Schilling Department of Mathematics, UC Davis based on joint work with Maria Gillespie, Graham Hawkes, Wencin Poh SageDays@ICERM: Combinatorics and Representation Theory July 23, 2018

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Goal Lie superalgebras: arose in physics to unify bosons and fermions

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Goal Lie superalgebras: arose in physics to unify bosons and fermions In mathematics: projective representations of the symmetric group

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Goal Lie superalgebras: arose in physics to unify bosons and fermions In mathematics: projective representations of the symmetric group Queer super Lie algebra

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Goal Lie superalgebras: arose in physics to unify bosons and fermions In mathematics: projective representations of the symmetric group Queer super Lie algebra Highest weight crystals for queer super Lie algebras (Grantcharov et al.)

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Goal Lie superalgebras: arose in physics to unify bosons and fermions In mathematics: projective representations of the symmetric group Queer super Lie algebra Highest weight crystals for queer super Lie algebras (Grantcharov et al.) Characterization of these crystals and how these discoveries were guided by experimentation with SageMath

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Outline Crystals of type A n 1 Queer supercrystals 2 Stembridge axioms 3 Characterization of queer crystals 4

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Crystals of type A n Abstract crystal of type A n : nonempty set B together with the maps e i , f i : B → B ⊔ { 0 } ( i ∈ I ) wt: B → Λ

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Crystals of type A n Abstract crystal of type A n : nonempty set B together with the maps e i , f i : B → B ⊔ { 0 } ( i ∈ I ) wt: B → Λ weight lattice Λ = Z n +1 � 0 index set I = { 1 , 2 , . . . , n } ǫ i i -th standard basis vector of Z n +1 simple root α i = ǫ i − ǫ i +1 ,

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Crystals of type A n Abstract crystal of type A n : nonempty set B together with the maps e i , f i : B → B ⊔ { 0 } ( i ∈ I ) wt: B → Λ weight lattice Λ = Z n +1 � 0 index set I = { 1 , 2 , . . . , n } ǫ i i -th standard basis vector of Z n +1 simple root α i = ǫ i − ǫ i +1 , string lengths for b ∈ B ϕ i ( b ) = max { k ∈ Z � 0 | f k i ( b ) � = 0 } ε i ( b ) = max { k ∈ Z � 0 | e k i ( b ) � = 0 }

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Crystals of type A n Abstract crystal of type A n : nonempty set B together with the maps e i , f i : B → B ⊔ { 0 } ( i ∈ I ) wt: B → Λ weight lattice Λ = Z n +1 � 0 index set I = { 1 , 2 , . . . , n } ǫ i i -th standard basis vector of Z n +1 simple root α i = ǫ i − ǫ i +1 , string lengths for b ∈ B ϕ i ( b ) = max { k ∈ Z � 0 | f k i ( b ) � = 0 } ε i ( b ) = max { k ∈ Z � 0 | e k i ( b ) � = 0 } We require: A1. f i b = b ′ if and only if b = e i b ′ wt( b ′ ) = wt( b ) + α i

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Crystal: A n example Example Standard crystal B for type A n : 1 2 3 n . . . 1 2 3 n + 1

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Crystal: A n example Example Standard crystal B for type A n : 1 2 3 n . . . 1 2 3 n + 1 � � wt i = ǫ i Highest weight element: 1

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Tensor products B and C crystals of type A n Definition Tensor product B ⊗ C has the following data:

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Tensor products B and C crystals of type A n Definition Tensor product B ⊗ C has the following data: Elements: b ⊗ c := ( b , c ) ∈ B × C

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Tensor products B and C crystals of type A n Definition Tensor product B ⊗ C has the following data: Elements: b ⊗ c := ( b , c ) ∈ B × C Weight map: wt( b ⊗ c ) = wt( b ) + wt( c )

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Tensor products B and C crystals of type A n Definition Tensor product B ⊗ C has the following data: Elements: b ⊗ c := ( b , c ) ∈ B × C Weight map: wt( b ⊗ c ) = wt( b ) + wt( c ) Crystal operators: � f i ( b ) ⊗ c if ε i ( b ) � ϕ i ( c ) f i ( b ⊗ c ) = b ⊗ f i ( c ) if ε i ( b ) < ϕ i ( c ) � e i ( b ) ⊗ c if ε i ( b ) > ϕ i ( c ) e i ( b ⊗ c ) = b ⊗ e i ( c ) if ε i ( b ) � ϕ i ( c )

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Example: Tensor product Example Components of crystal of words B ⊗ 3 = B ⊗ B ⊗ B of type A 2 : 1 ⊗ 2 ⊗ 1 1 ⊗ 1 ⊗ 1 2 ⊗ 1 ⊗ 1 3 ⊗ 2 ⊗ 1 2 1 1 2 1 1 ⊗ 3 ⊗ 1 2 ⊗ 2 ⊗ 1 1 ⊗ 1 ⊗ 2 3 ⊗ 1 ⊗ 1 2 ⊗ 1 ⊗ 2 1 2 1 2 1 2 1 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 1 1 ⊗ 2 ⊗ 2 1 ⊗ 1 ⊗ 3 3 ⊗ 1 ⊗ 2 2 ⊗ 1 ⊗ 3 1 2 1 2 1 1 2 2 ⊗ 3 ⊗ 2 3 ⊗ 3 ⊗ 1 2 ⊗ 2 ⊗ 2 1 ⊗ 2 ⊗ 3 3 ⊗ 2 ⊗ 2 3 ⊗ 1 ⊗ 3 2 1 2 1 2 2 1 3 ⊗ 3 ⊗ 2 2 ⊗ 2 ⊗ 3 1 ⊗ 3 ⊗ 3 3 ⊗ 2 ⊗ 3 2 1 2 ⊗ 3 ⊗ 3 2 3 ⊗ 3 ⊗ 3

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Motivation Why are crystals interesting?

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Motivation Why are crystals interesting? Characters: character of highest weight crystal B ( λ ) is Schur function s λ

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Motivation Why are crystals interesting? Characters: character of highest weight crystal B ( λ ) is Schur function s λ Littlewood–Richardson rule: � c ν s λ s µ = λµ s ν ν c ν λµ = number of highest weights of weight ν in B ( λ ) ⊗ B ( µ )

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Outline Crystals of type A n 1 Queer supercrystals 2 Stembridge axioms 3 Characterization of queer crystals 4

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Queer crystal: Developments Queer Lie superalgebra q ( n ): a super analogue of gl ( n )

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Queer crystal: Developments Queer Lie superalgebra q ( n ): a super analogue of gl ( n ) [Grantcharov, Jung, Kang, Kashiwara, Kim, ’10]: Crystal basis theory for queer Lie superalgebras using U q ( q ( n )) ◮ Introduced queer crystals on words with tensor product rule. ◮ Explicit combinatorial realization of queer crystals using semistandard decomposition tableaux. ◮ Existence of fake highest (and lowest) weights on queer crystals.

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Standard crystal and tensor product Example Standard queer crystal B for q ( n + 1) 1 2 3 n . . . 1 2 3 n + 1 − 1

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Standard crystal and tensor product Example Standard queer crystal B for q ( n + 1) 1 2 3 n . . . 1 2 3 n + 1 − 1 Tensor product: b ⊗ c ∈ B ⊗ C � b ⊗ f − 1 ( c ) if wt( b ) 1 = wt( b ) 2 = 0 f − 1 ( b ⊗ c ) = f − 1 ( b ) ⊗ c otherwise � b ⊗ e − 1 ( c ) if wt( b ) 1 = wt( b ) 2 = 0 e − 1 ( b ⊗ c ) = e − 1 ( b ) ⊗ c otherwise

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Queer crystal: Example One connected component of B ⊗ 4 for q (3): 1 ⊗ 1 ⊗ 2 ⊗ 1 2 1 1 2 ⊗ 1 ⊗ 2 ⊗ 1 1 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 2 ⊗ 2 ⊗ 1 2 1 2 1 1 1 3 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 1 ⊗ 3 ⊗ 2 1 ⊗ 2 ⊗ 3 ⊗ 1 2 ⊗ 2 ⊗ 2 ⊗ 1 1 2 1 1 2 1 2 1 1 1 3 ⊗ 2 ⊗ 2 ⊗ 1 3 ⊗ 1 ⊗ 3 ⊗ 1 2 ⊗ 1 ⊗ 3 ⊗ 2 1 ⊗ 2 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 3 ⊗ 1 2 ⊗ 2 ⊗ 3 ⊗ 1 2 1 2 1 2 1 2 1 1 1 3 ⊗ 2 ⊗ 3 ⊗ 1 3 ⊗ 1 ⊗ 3 ⊗ 2 2 ⊗ 2 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 3 ⊗ 1 1 2 1 2 1 1 3 ⊗ 2 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 3 ⊗ 2 3 ⊗ 3 ⊗ 3 ⊗ 1 2 1 1 3 ⊗ 3 ⊗ 3 ⊗ 2

Crystals of type A n Queer supercrystals Stembridge axioms Characterization of queer crystals Motivation Why are queer crystals interesting? Characters: character of highest weight crystal B ( λ ) ( λ strict partition) is Schur P function P λ