Characterization of queer supercrystals Maria Gillespie, UC Davis - PowerPoint PPT Presentation

Characterization of queer supercrystals Maria Gillespie, UC Davis On joint work with Graham Hawkes, Wencin Poh, and Anne Schilling CanaDAM, Minisymposium on Algebraic and Geometric Methods in Combinatorics May 30, 2019

✏ ➓ ➓ ♣ q Why ‘Crystals’? ➓ Crystals arise at cold temperatures! ➓ Kashiwara: ‘crystal bases’ of representations of quantum groups U q ♣ g q in the limit q Ñ 0 ( q is temperature). ➓ Rigid combinatorial structures with applications to symmetric function theory, representation theory, geometry...

Why ‘Crystals’? ➓ Crystals arise at cold temperatures! ➓ Kashiwara: ‘crystal bases’ of representations of quantum groups U q ♣ g q in the limit q Ñ 0 ( q is temperature). ➓ Rigid combinatorial structures with applications to symmetric function theory, representation theory, geometry... Talk outline: ➓ Part 1: Type A crystals (for Lie algebra g ✏ sl n ) ➓ Part 2: Queer supercrystals (for quantum queer Lie superalgebra q ♣ n q )

✁ ④♣ q ✏ ♣ ✁ q ✏ ✁ � ✂ ✡ ✂ ✡ ✂ ✡ ✁ ♣ q ❜ ✁ ❜ ✏ r s ♣ q ✏ ♣ q Ñ Ñ Lie algebras: Notation and Background Notation Example/Description Lie algebra g sl n (trace-0 n ✂ n matrices) Lie bracket r , s r x , y s ✏ xy ✁ yx Classical types: A n , B n , C n , D n Weight lattice Λ Simple roots α i , i P I Generators e i , f i , h i Univ. envel. alg. U ♣ g q Quantized UEA U q ♣ g q

④♣ q ✏ ♣ ✁ q ✏ ✁ � ✂ ✡ ✂ ✡ ✂ ✡ ✁ ♣ q ❜ ✁ ❜ ✏ r s ♣ q ✏ ♣ q Ñ Ñ Lie algebras: Notation and Background Notation Example/Description Lie algebra g sl n (trace-0 n ✂ n matrices) Lie bracket r , s r x , y s ✏ xy ✁ yx Classical types: Type A n ✁ 1 A n , B n , C n , D n Weight lattice Λ Simple roots α i , i P I Generators e i , f i , h i Univ. envel. alg. U ♣ g q Quantized UEA U q ♣ g q

✏ ♣ ✁ q ✏ ✁ � ✂ ✡ ✂ ✡ ✂ ✡ ✁ ♣ q ❜ ✁ ❜ ✏ r s ♣ q ✏ ♣ q Ñ Ñ Lie algebras: Notation and Background Notation Example/Description Lie algebra g sl n (trace-0 n ✂ n matrices) Lie bracket r , s r x , y s ✏ xy ✁ yx Classical types: Type A n ✁ 1 A n , B n , C n , D n Z n ④♣ 1 , 1 , . . . , 1 q Weight lattice Λ Simple roots α i , i P I Generators e i , f i , h i Univ. envel. alg. U ♣ g q Quantized UEA U q ♣ g q

✂ ✡ ✂ ✡ ✂ ✡ ✁ ♣ q ❜ ✁ ❜ ✏ r s ♣ q ✏ ♣ q Ñ Ñ Lie algebras: Notation and Background Notation Example/Description Lie algebra g sl n (trace-0 n ✂ n matrices) Lie bracket r , s r x , y s ✏ xy ✁ yx Classical types: Type A n ✁ 1 A n , B n , C n , D n Z n ④♣ 1 , 1 , . . . , 1 q Weight lattice Λ Simple roots α i , i P I α i ✏ ♣ 0 , . . . , 0 , 1 , ✁ 1 , 0 , . . . , 0 q ✏ e i ✁ e i � 1 Generators e i , f i , h i Univ. envel. alg. U ♣ g q Quantized UEA U q ♣ g q

♣ q ❜ ✁ ❜ ✏ r s ♣ q ✏ ♣ q Ñ Ñ Lie algebras: Notation and Background Notation Example/Description Lie algebra g sl n (trace-0 n ✂ n matrices) Lie bracket r , s r x , y s ✏ xy ✁ yx Classical types: Type A n ✁ 1 A n , B n , C n , D n Z n ④♣ 1 , 1 , . . . , 1 q Weight lattice Λ Simple roots α i , i P I α i ✏ ♣ 0 , . . . , 0 , 1 , ✁ 1 , 0 , . . . , 0 q ✏ e i ✁ e i � 1 ✂ 0 ✂ 0 ✂ 1 1 ✡ 0 ✡ 0 ✡ Generators e i , f i , h i for sl 2 , , 0 0 1 0 0 ✁ 1 (Raising, lowering, wt-preserving) Univ. envel. alg. U ♣ g q Quantized UEA U q ♣ g q

♣ q ✏ ♣ q Ñ Ñ Lie algebras: Notation and Background Notation Example/Description Lie algebra g sl n (trace-0 n ✂ n matrices) Lie bracket r , s r x , y s ✏ xy ✁ yx Classical types: Type A n ✁ 1 A n , B n , C n , D n Z n ④♣ 1 , 1 , . . . , 1 q Weight lattice Λ Simple roots α i , i P I α i ✏ ♣ 0 , . . . , 0 , 1 , ✁ 1 , 0 , . . . , 0 q ✏ e i ✁ e i � 1 ✂ 0 ✂ 0 ✂ 1 1 ✡ 0 ✡ 0 ✡ Generators e i , f i , h i for sl 2 , , 0 0 1 0 0 ✁ 1 (Raising, lowering, wt-preserving) Univ. envel. alg. U ♣ g q T ♣ g q mod x ❜ y ✁ y ❜ x ✏ r x , y s Contains all g -reps; gen. by e i , f i , h i Quantized UEA U q ♣ g q

Lie algebras: Notation and Background Notation Example/Description Lie algebra g sl n (trace-0 n ✂ n matrices) Lie bracket r , s r x , y s ✏ xy ✁ yx Classical types: Type A n ✁ 1 A n , B n , C n , D n Z n ④♣ 1 , 1 , . . . , 1 q Weight lattice Λ Simple roots α i , i P I α i ✏ ♣ 0 , . . . , 0 , 1 , ✁ 1 , 0 , . . . , 0 q ✏ e i ✁ e i � 1 ✂ 0 ✂ 0 ✂ 1 1 ✡ 0 ✡ 0 ✡ Generators e i , f i , h i for sl 2 , , 0 0 1 0 0 ✁ 1 (Raising, lowering, wt-preserving) Univ. envel. alg. U ♣ g q T ♣ g q mod x ❜ y ✁ y ❜ x ✏ r x , y s Contains all g -reps; gen. by e i , f i , h i Quantized UEA U q ♣ g q lim q Ñ 1 U q ♣ g q ✏ U ♣ g q q Ñ 0: crystal bases for reps

Ñ ➓ ✏ ④♣ q Ñ ❨ t ✉ ➓ ♣ ♣ qq ✏ ♣ q ✁ ✏ ♣ ✁ q ✏ ♣ ✁ q Ñ ❨ t ✉ ➓ Ñ ➓ ♣ q ✏ t ♣ q ✘ ✉ ♣ q ✏ t ♣ q ✘ ✉ Lie algebra crystals (Ex. g ✏ sl 3 ) (2 , 1 , 0) ➓ Ground set B (“base”) (1 , 2 , 0) (2 , 0 , 1) (1 , 1 , 1) (0 , 2 , 1) (1 , 0 , 2) (0 , 1 , 2)

Ñ ❨ t ✉ ➓ ♣ ♣ qq ✏ ♣ q ✁ ✏ ♣ ✁ q ✏ ♣ ✁ q Ñ ❨ t ✉ ➓ Ñ ➓ ♣ q ✏ t ♣ q ✘ ✉ ♣ q ✏ t ♣ q ✘ ✉ Lie algebra crystals (Ex. g ✏ sl 3 ) (2 , 1 , 0) ➓ Ground set B (“base”) ➓ Weight map wt : B Ñ Λ (Ex. Λ ✏ Z 3 ④♣ 1 , 1 , 1 q ) (1 , 2 , 0) (2 , 0 , 1) (1 , 1 , 1) (0 , 2 , 1) (1 , 0 , 2) (0 , 1 , 2)

Ñ ❨ t ✉ ➓ Ñ ➓ ♣ q ✏ t ♣ q ✘ ✉ ♣ q ✏ t ♣ q ✘ ✉ Lie algebra crystals (Ex. g ✏ sl 3 ) (2 , 1 , 0) ➓ Ground set B (“base”) ➓ Weight map wt : B Ñ Λ (Ex. f 1 Λ ✏ Z 3 ④♣ 1 , 1 , 1 q ) (1 , 2 , 0) ➓ Operators f i : B Ñ B ❨ t 0 ✉ , wt ♣ f i ♣ x qq ✏ wt ♣ x q ✁ α i ( α 1 ✏ ♣ 1 , ✁ 1 , 0 q , α 2 ✏ ♣ 0 , 1 , ✁ 1 q )

Ñ ❨ t ✉ ➓ Ñ ➓ ♣ q ✏ t ♣ q ✘ ✉ ♣ q ✏ t ♣ q ✘ ✉ Lie algebra crystals (Ex. g ✏ sl 3 ) (2 , 1 , 0) ➓ Ground set B (“base”) ➓ Weight map wt : B Ñ Λ (Ex. f 1 f 2 Λ ✏ Z 3 ④♣ 1 , 1 , 1 q ) (2 , 0 , 1) ➓ Operators f i : B Ñ B ❨ t 0 ✉ , wt ♣ f i ♣ x qq ✏ wt ♣ x q ✁ α i ( α 1 ✏ ♣ 1 , ✁ 1 , 0 q , α 2 ✏ ♣ 0 , 1 , ✁ 1 q )

Ñ ➓ ♣ q ✏ t ♣ q ✘ ✉ ♣ q ✏ t ♣ q ✘ ✉ Lie algebra crystals (Ex. g ✏ sl 3 ) (2 , 1 , 0) ➓ Ground set B (“base”) ➓ Weight map wt : B Ñ Λ (Ex. Λ ✏ Z 3 ④♣ 1 , 1 , 1 q ) ➓ Operators f i : B Ñ B ❨ t 0 ✉ , wt ♣ f i ♣ x qq ✏ wt ♣ x q ✁ α i ( α 1 ✏ ♣ 1 , ✁ 1 , 0 q , α 2 ✏ ♣ 0 , 1 , ✁ 1 q ) ➓ Operators e i : B Ñ B ❨ t 0 ✉ partial inverse of f i

Lie algebra crystals (Ex. g ✏ sl 3 ) ϕ 1 = ϕ 2 = 1 ➓ Ground set B (“base”) ➓ Weight map wt : B Ñ Λ (Ex. Λ ✏ Z 3 ④♣ 1 , 1 , 1 q ) ➓ Operators f i : B Ñ B ❨ t 0 ✉ , wt ♣ f i ♣ x qq ✏ wt ♣ x q ✁ α i ( α 1 ✏ ♣ 1 , ✁ 1 , 0 q , α 2 ✏ ♣ 0 , 1 , ✁ 1 q ) ➓ Operators e i : B Ñ B ❨ t 0 ✉ partial inverse of f i ➓ Lengths ϕ i , ε i : B Ñ Z , usually: ϕ i ♣ x q ✏ max t k : f k i ♣ x q ✘ 0 ✉ ε i ♣ x q ✏ max t k : e k i ♣ x q ✘ 0 ✉

� ✟ ♣ q ✏ ♣ ♣ q ✁ ♣ q ♣ q ✁ ♣ qq ✏ ♣ ✁ q ♣ q ⑤ ✁ ⑤ ➙ ♣ q ✏ � ♣ q ✏ ✏ ♣ � ♣ q ✁ � ♣ q ♣ q ✁ ♣ qq ✘ ♣ q ✏ ♣ q Stembridge crystals ➓ Stembridge : ‘Local axioms’ determine which crystals correspond to U q ♣ g q -representations (for simply-laced types). ➓ Lengths Axiom: ➓ Non-adjacent operators: ➓ Adjacent operators:

� ⑤ ✁ ⑤ ➙ ♣ q ✏ � ♣ q ✏ ✏ ♣ � ♣ q ✁ � ♣ q ♣ q ✁ ♣ qq ✘ ♣ q ✏ ♣ q Stembridge crystals ➓ Stembridge : ‘Local axioms’ determine which crystals correspond to U q ♣ g q -representations (for simply-laced types). ➓ Lengths Axiom: If f i ✟ 1 ♣ w q ✏ x , then ♣ ε i ♣ w q ✁ ε i ♣ x q , ϕ i ♣ w q ✁ ϕ i ♣ x qq ✏ ♣ 0 , ✁ 1 q or ♣ 1 , 0 q . ➓ Non-adjacent operators: ➓ Adjacent operators:

� ♣ q ✏ � ♣ q ✏ ✏ ♣ � ♣ q ✁ � ♣ q ♣ q ✁ ♣ qq ✘ ♣ q ✏ ♣ q Stembridge crystals ➓ Stembridge : ‘Local axioms’ determine which crystals correspond to U q ♣ g q -representations (for simply-laced types). ➓ Lengths Axiom: If f i ✟ 1 ♣ w q ✏ x , then ♣ ε i ♣ w q ✁ ε i ♣ x q , ϕ i ♣ w q ✁ ϕ i ♣ x qq ✏ ♣ 0 , ✁ 1 q or ♣ 1 , 0 q . ➓ Non-adjacent operators: If ⑤ i ✁ j ⑤ ➙ 2 then f i , f j commute. ➓ Adjacent operators:

Stembridge crystals ➓ Stembridge : ‘Local axioms’ determine which crystals correspond to U q ♣ g q -representations (for simply-laced types). ➓ Lengths Axiom: If f i ✟ 1 ♣ w q ✏ x , then ♣ ε i ♣ w q ✁ ε i ♣ x q , ϕ i ♣ w q ✁ ϕ i ♣ x qq ✏ ♣ 0 , ✁ 1 q or ♣ 1 , 0 q . ➓ Non-adjacent operators: If ⑤ i ✁ j ⑤ ➙ 2 then f i , f j commute. ➓ Adjacent operators: Suppose f i ♣ w q ✏ x and f i � 1 ♣ w q ✏ y . Define ∆ : ✏ ♣ ε i � 1 ♣ w q ✁ ε i � 1 ♣ x q , ε i ♣ w q ✁ ε i ♣ y qq . Then: ∆ ✘ ♣ 0 , 0 q ∆ ✏ ♣ 0 , 0 q • • • • (And dual statements • • • • for e i , e i � 1 .) • • • •