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

Why ‘Crystals’?

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

Why ‘Crystals’?

➓ Crystals arise at cold temperatures! ➓ Kashiwara: ‘crystal bases’ of representations of quantum groups Uq♣gq 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 ✏ sln) ➓ Part 2: Queer supercrystals (for quantum queer Lie superalgebra q♣nq)

Lie algebras: Notation and Background

Notation Example/Description Lie algebra g sln (trace-0 n ✂ n matrices) Lie bracket r, s rx, ys ✏ xy ✁ yx Classical types:

✁

An, Bn, Cn, Dn Weight lattice Λ ④♣ q Simple roots αi, i P I ✏ ♣ ✁ q ✏ ✁

• Generators ei, fi, hi

✂ ✡ ✂ ✡ ✂ ✁ ✡

• Univ. envel. alg. U♣gq

♣ q ❜ ✁ ❜ ✏ r s Quantized UEA Uq♣gq

Ñ

♣ q ✏ ♣ q Ñ

Lie algebras: Notation and Background

Notation Example/Description Lie algebra g sln (trace-0 n ✂ n matrices) Lie bracket r, s rx, ys ✏ xy ✁ yx Classical types: Type An✁1 An, Bn, Cn, Dn Weight lattice Λ ④♣ q Simple roots αi, i P I ✏ ♣ ✁ q ✏ ✁

• Generators ei, fi, hi

✂ ✡ ✂ ✡ ✂ ✁ ✡

• Univ. envel. alg. U♣gq

♣ q ❜ ✁ ❜ ✏ r s Quantized UEA Uq♣gq

Ñ

♣ q ✏ ♣ q Ñ

Lie algebras: Notation and Background

Notation Example/Description Lie algebra g sln (trace-0 n ✂ n matrices) Lie bracket r, s rx, ys ✏ xy ✁ yx Classical types: Type An✁1 An, Bn, Cn, Dn Weight lattice Λ Zn④♣1, 1, . . . , 1q Simple roots αi, i P I ✏ ♣ ✁ q ✏ ✁

• Generators ei, fi, hi

✂ ✡ ✂ ✡ ✂ ✁ ✡

• Univ. envel. alg. U♣gq

♣ q ❜ ✁ ❜ ✏ r s Quantized UEA Uq♣gq

Ñ

♣ q ✏ ♣ q Ñ

Lie algebras: Notation and Background

Notation Example/Description Lie algebra g sln (trace-0 n ✂ n matrices) Lie bracket r, s rx, ys ✏ xy ✁ yx Classical types: Type An✁1 An, Bn, Cn, Dn Weight lattice Λ Zn④♣1, 1, . . . , 1q Simple roots αi, i P I αi ✏ ♣0, . . . , 0, 1, ✁1, 0, . . . , 0q ✏ ei ✁ ei1 Generators ei, fi, hi ✂ ✡ ✂ ✡ ✂ ✁ ✡

• Univ. envel. alg. U♣gq

♣ q ❜ ✁ ❜ ✏ r s Quantized UEA Uq♣gq

Ñ

♣ q ✏ ♣ q Ñ

Lie algebras: Notation and Background

Notation Example/Description Lie algebra g sln (trace-0 n ✂ n matrices) Lie bracket r, s rx, ys ✏ xy ✁ yx Classical types: Type An✁1 An, Bn, Cn, Dn Weight lattice Λ Zn④♣1, 1, . . . , 1q Simple roots αi, i P I αi ✏ ♣0, . . . , 0, 1, ✁1, 0, . . . , 0q ✏ ei ✁ ei1 Generators ei, fi, hi ✂ 0 1 ✡ , ✂ 0 1 ✡ , ✂ 1 ✁1 ✡ for sl2 (Raising, lowering, wt-preserving)

• Univ. envel. alg. U♣gq

♣ q ❜ ✁ ❜ ✏ r s Quantized UEA Uq♣gq

Ñ

♣ q ✏ ♣ q Ñ

Lie algebras: Notation and Background

Notation Example/Description Lie algebra g sln (trace-0 n ✂ n matrices) Lie bracket r, s rx, ys ✏ xy ✁ yx Classical types: Type An✁1 An, Bn, Cn, Dn Weight lattice Λ Zn④♣1, 1, . . . , 1q Simple roots αi, i P I αi ✏ ♣0, . . . , 0, 1, ✁1, 0, . . . , 0q ✏ ei ✁ ei1 Generators ei, fi, hi ✂ 0 1 ✡ , ✂ 0 1 ✡ , ✂ 1 ✁1 ✡ for sl2 (Raising, lowering, wt-preserving)

• Univ. envel. alg. U♣gq

T♣gq mod x ❜ y ✁ y ❜ x ✏ rx, ys Contains all g-reps; gen. by ei, fi, hi Quantized UEA Uq♣gq

Ñ

♣ q ✏ ♣ q Ñ

Lie algebras: Notation and Background

Notation Example/Description Lie algebra g sln (trace-0 n ✂ n matrices) Lie bracket r, s rx, ys ✏ xy ✁ yx Classical types: Type An✁1 An, Bn, Cn, Dn Weight lattice Λ Zn④♣1, 1, . . . , 1q Simple roots αi, i P I αi ✏ ♣0, . . . , 0, 1, ✁1, 0, . . . , 0q ✏ ei ✁ ei1 Generators ei, fi, hi ✂ 0 1 ✡ , ✂ 0 1 ✡ , ✂ 1 ✁1 ✡ for sl2 (Raising, lowering, wt-preserving)

• Univ. envel. alg. U♣gq

T♣gq mod x ❜ y ✁ y ❜ x ✏ rx, ys Contains all g-reps; gen. by ei, fi, hi Quantized UEA Uq♣gq limqÑ1 Uq♣gq ✏ U♣gq q Ñ 0: crystal bases for reps

Lie algebra crystals

(Ex. g ✏ sl3) ➓ Ground set B (“base”) ➓ Ñ ✏ ④♣ q ➓ Ñ ❨ t ✉ ♣ ♣ qq ✏ ♣ q ✁ ✏ ♣ ✁ q ✏ ♣ ✁ q ➓ Ñ ❨ t ✉ ➓ Ñ ♣ q ✏ t ♣ q ✘ ✉ ♣ q ✏ t ♣ q ✘ ✉

(2, 1, 0) (1, 2, 0) (2, 0, 1) (1, 1, 1) (0, 2, 1) (1, 0, 2) (0, 1, 2)

Lie algebra crystals

(Ex. g ✏ sl3) ➓ Ground set B (“base”) ➓ Weight map wt : B Ñ Λ (Ex. Λ ✏ Z3④♣1, 1, 1q) ➓ Ñ ❨ t ✉ ♣ ♣ qq ✏ ♣ q ✁ ✏ ♣ ✁ q ✏ ♣ ✁ q ➓ Ñ ❨ t ✉ ➓ Ñ ♣ q ✏ t ♣ q ✘ ✉ ♣ q ✏ t ♣ q ✘ ✉

(2, 1, 0) (1, 2, 0) (2, 0, 1) (1, 1, 1) (0, 2, 1) (1, 0, 2) (0, 1, 2)

Lie algebra crystals

(Ex. g ✏ sl3) ➓ Ground set B (“base”) ➓ Weight map wt : B Ñ Λ (Ex. Λ ✏ Z3④♣1, 1, 1q) ➓ Operators fi : B Ñ B ❨ t0✉, wt♣fi♣xqq ✏ wt♣xq ✁ αi (α1 ✏ ♣1, ✁1, 0q, α2 ✏ ♣0, 1, ✁1q) ➓ Ñ ❨ t ✉ ➓ Ñ ♣ q ✏ t ♣ q ✘ ✉ ♣ q ✏ t ♣ q ✘ ✉

(2, 1, 0) (1, 2, 0) f1

Lie algebra crystals

(Ex. g ✏ sl3) ➓ Ground set B (“base”) ➓ Weight map wt : B Ñ Λ (Ex. Λ ✏ Z3④♣1, 1, 1q) ➓ Operators fi : B Ñ B ❨ t0✉, wt♣fi♣xqq ✏ wt♣xq ✁ αi (α1 ✏ ♣1, ✁1, 0q, α2 ✏ ♣0, 1, ✁1q) ➓ Ñ ❨ t ✉ ➓ Ñ ♣ q ✏ t ♣ q ✘ ✉ ♣ q ✏ t ♣ q ✘ ✉

(2, 1, 0) (2, 0, 1) f1 f2

Lie algebra crystals

(Ex. g ✏ sl3) ➓ Ground set B (“base”) ➓ Weight map wt : B Ñ Λ (Ex. Λ ✏ Z3④♣1, 1, 1q) ➓ Operators fi : B Ñ B ❨ t0✉, wt♣fi♣xqq ✏ wt♣xq ✁ αi (α1 ✏ ♣1, ✁1, 0q, α2 ✏ ♣0, 1, ✁1q) ➓ Operators ei : B Ñ B ❨ t0✉ partial inverse of fi ➓ Ñ ♣ q ✏ t ♣ q ✘ ✉ ♣ q ✏ t ♣ q ✘ ✉

(2, 1, 0)

Lie algebra crystals

(Ex. g ✏ sl3) ➓ Ground set B (“base”) ➓ Weight map wt : B Ñ Λ (Ex. Λ ✏ Z3④♣1, 1, 1q) ➓ Operators fi : B Ñ B ❨ t0✉, wt♣fi♣xqq ✏ wt♣xq ✁ αi (α1 ✏ ♣1, ✁1, 0q, α2 ✏ ♣0, 1, ✁1q) ➓ Operators ei : B Ñ B ❨ t0✉ partial inverse of fi ➓ Lengths ϕi, εi : B Ñ Z, usually: ϕi♣xq ✏ maxtk : f k

i ♣xq ✘ 0✉

εi♣xq ✏ maxtk : ek

i ♣xq ✘ 0✉

ϕ1 = ϕ2 = 1

Stembridge crystals

➓ Stembridge: ‘Local axioms’ determine which crystals correspond to Uq♣gq-representations (for simply-laced types).

➓ Lengths Axiom:

✟ ♣ q ✏

♣ ♣ q ✁ ♣ q ♣ q ✁ ♣ qq ✏ ♣ ✁ q ♣ q ➓ Non-adjacent operators: ⑤ ✁ ⑤ ➙ ➓ Adjacent operators: ♣ q ✏

♣ q ✏

✏ ♣

♣ q ✁ ♣ q

♣ q ✁ ♣ qq ✘ ♣ q ✏ ♣ q

Stembridge crystals

➓ Stembridge: ‘Local axioms’ determine which crystals correspond to Uq♣gq-representations (for simply-laced types).

➓ Lengths Axiom: If fi✟1♣wq ✏ x, then ♣εi♣wq ✁ εi♣xq, ϕi♣wq ✁ ϕi♣xqq ✏ ♣0, ✁1q or ♣1, 0q. ➓ Non-adjacent operators: ⑤ ✁ ⑤ ➙ ➓ Adjacent operators: ♣ q ✏

♣ q ✏

✏ ♣

♣ q ✁ ♣ q

♣ q ✁ ♣ qq ✘ ♣ q ✏ ♣ q

Stembridge crystals

➓ Stembridge: ‘Local axioms’ determine which crystals correspond to Uq♣gq-representations (for simply-laced types).

➓ Lengths Axiom: If fi✟1♣wq ✏ x, then ♣εi♣wq ✁ εi♣xq, ϕi♣wq ✁ ϕi♣xqq ✏ ♣0, ✁1q or ♣1, 0q. ➓ Non-adjacent operators: If ⑤i ✁ j⑤ ➙ 2 then fi, fj commute. ➓ Adjacent operators: ♣ q ✏

♣ q ✏

✏ ♣

♣ q ✁ ♣ q

♣ q ✁ ♣ qq ✘ ♣ q ✏ ♣ q

Stembridge crystals

➓ Stembridge: ‘Local axioms’ determine which crystals correspond to Uq♣gq-representations (for simply-laced types).

➓ Lengths Axiom: If fi✟1♣wq ✏ x, then ♣εi♣wq ✁ εi♣xq, ϕi♣wq ✁ ϕi♣xqq ✏ ♣0, ✁1q or ♣1, 0q. ➓ Non-adjacent operators: If ⑤i ✁ j⑤ ➙ 2 then fi, fj commute. ➓ Adjacent operators: Suppose fi♣wq ✏ x and fi1♣wq ✏ y. Define ∆ :✏ ♣εi1♣wq ✁ εi1♣xq, εi♣wq ✁ εi♣yqq. Then: ∆ ✘ ♣0, 0q ∆ ✏ ♣0, 0q

• (And dual

statements for ei, ei1.)

Stembridge crystals

➓ Stembridge: ‘Local axioms’ determine which crystals correspond to Uq♣gq-representations (for simply-laced types).

➓ Lengths Axiom: If fi✟1♣wq ✏ x, then ♣εi♣wq ✁ εi♣xq, ϕi♣wq ✁ ϕi♣xqq ✏ ♣0, ✁1q or ♣1, 0q. ➓ Non-adjacent operators: If ⑤i ✁ j⑤ ➙ 2 then fi, fj commute. ➓ Adjacent operators: Suppose fi♣wq ✏ x and fi1♣wq ✏ y. Define ∆ :✏ ♣εi1♣wq ✁ εi1♣xq, εi♣wq ✁ εi♣yqq. Then: ∆ ✘ ♣0, 0q ∆ ✏ ♣0, 0q

• (And dual

statements for ei, ei1.)

Stembridge crystals

➓ Stembridge: ‘Local axioms’ determine which crystals correspond to Uq♣gq-representations (for simply-laced types).

➓ Lengths Axiom: If fi✟1♣wq ✏ x, then ♣εi♣wq ✁ εi♣xq, ϕi♣wq ✁ ϕi♣xqq ✏ ♣0, ✁1q or ♣1, 0q. ➓ Non-adjacent operators: If ⑤i ✁ j⑤ ➙ 2 then fi, fj commute. ➓ Adjacent operators: Suppose fi♣wq ✏ x and fi1♣wq ✏ y. Define ∆ :✏ ♣εi1♣wq ✁ εi1♣xq, εi♣wq ✁ εi♣yqq. Then: ∆ ✘ ♣0, 0q ∆ ✏ ♣0, 0q

• (And dual

statements for ei, ei1.)

Properties of Stembridge crystals

➓ Unique highest weight elements (killed by all ei operators) ➓ ➓ ➳

P ♣ q

➓ ✏ ➳

(2, 1, 0) (2, 0, 1) f1 f2

Properties of Stembridge crystals

➓ Unique highest weight elements (killed by all ei operators) ➓ Component determined uniquely by its highest weight ➓ ➳

P ♣ q

➓ ✏ ➳

(2, 1, 0) (2, 0, 1) f1 f2

Properties of Stembridge crystals

➓ Unique highest weight elements (killed by all ei operators) ➓ Component determined uniquely by its highest weight ➓ In type A: if highest weight is partition λ, character ➳

bPB

xwt♣bq is Schur function sλ ➓ ✏ ➳

(2, 1, 0) (2, 0, 1) f1 f2

Properties of Stembridge crystals

➓ Unique highest weight elements (killed by all ei operators) ➓ Component determined uniquely by its highest weight ➓ In type A: if highest weight is partition λ, character ➳

bPB

xwt♣bq is Schur function sλ ➓ Can recover Littlewood-Richardson rule: sλsµ ✏ ➳ cν

λµsν

via crystal tensor products

(2, 1, 0) (2, 0, 1) f1 f2

Tensor products of crystals

Tensor product B ❜ C is the crystal having: ➓ Ground set B ✂ C ➓ Weight function wt♣x ❜ yq ✏ wt♣xq wt♣yq ➓ Operators ei♣x ❜ yq ✏ ★ ei♣xq ❜ y ϕi♣yq ➔ εi♣xq x ❜ ei♣yq ϕi♣yq ➙ εi♣xq fi♣x ❜ yq ✏ ★ fi♣xq ❜ y ϕi♣yq ↕ εi♣xq x ❜ fi♣yq ϕi♣yq → εi♣xq

Standard crystal and tensor products

Standard crystal B0 for sln: 1 2 3 . . . n 1 2 3 n ✁ 1 Components of crystal of words B❜3 ✏ B0 ❜ B0 ❜ B0 for sl3:

3 ❜ 3 ❜ 3 3 ❜ 3 ❜ 1 3 ❜ 3 ❜ 2 3 ❜ 2 ❜ 2 3 ❜ 2 ❜ 3 3 ❜ 1 ❜ 3 3 ❜ 2 ❜ 1 2 ❜ 1 ❜ 1 1 ❜ 3 ❜ 3 1 ❜ 3 ❜ 2 1 ❜ 3 ❜ 1 1 ❜ 2 ❜ 3 1 ❜ 2 ❜ 2 1 ❜ 2 ❜ 1 1 ❜ 1 ❜ 3 1 ❜ 1 ❜ 2 1 ❜ 1 ❜ 1 3 ❜ 1 ❜ 2 3 ❜ 1 ❜ 1 2 ❜ 3 ❜ 3 2 ❜ 3 ❜ 2 2 ❜ 3 ❜ 1 2 ❜ 2 ❜ 3 2 ❜ 2 ❜ 2 2 ❜ 2 ❜ 1 2 ❜ 1 ❜ 3 2 ❜ 1 ❜ 2 2 1 1 2 2 1 1 2 2 2 2 1 2 1 1 1 1 1 2 1 2 2 1 2 1 2 1 2

Part 2: Lie superalgebras and q♣nq

➓ Lie superalgebra: Z2-graded algebra g0 ❵ g1 with ‘super’ Lie bracket r, s. Example: rx, ys ✏ ★ xy ✁ yx x P g0 or y P g0 xy yx x, y P g1 ➓

➓ ♣ q ♣ q ♣ q ♣ q ➓ ♣ q ➓ ♣ q ♣ q ➓ ♣ q ♣ q

➓ ♣ q ♣ q ✕ ❵ ♣ q

✁ ✁ ✁

♣ q

Part 2: Lie superalgebras and q♣nq

➓ Lie superalgebra: Z2-graded algebra g0 ❵ g1 with ‘super’ Lie bracket r, s. Example: rx, ys ✏ ★ xy ✁ yx x P g0 or y P g0 xy yx x, y P g1 ➓ ‘Classical’ Lie superalgebras (simple, g1 is reducible g0-rep):

➓ Main series: A♣m, nq, B♣m, nq, C♣nq, D♣m, nq ➓ Deformations: D♣2, 1; αq ➓ Exceptional: G♣3q, F♣4q ➓ Strange: P♣nq, Q♣nq (also analog of type A Lie algebra)

➓ ♣ q ♣ q ✕ ❵ ♣ q

✁ ✁ ✁

♣ q

Part 2: Lie superalgebras and q♣nq

➓ Lie superalgebra: Z2-graded algebra g0 ❵ g1 with ‘super’ Lie bracket r, s. Example: rx, ys ✏ ★ xy ✁ yx x P g0 or y P g0 xy yx x, y P g1 ➓ ‘Classical’ Lie superalgebras (simple, g1 is reducible g0-rep):

➓ Main series: A♣m, nq, B♣m, nq, C♣nq, D♣m, nq ➓ Deformations: D♣2, 1; αq ➓ Exceptional: G♣3q, F♣4q ➓ Strange: P♣nq, Q♣nq (also analog of type A Lie algebra)

➓ Type Q♣nq: queer Lie superalgebra q♣nq ✕ sln ❵ sln, generators ei, fi, hi for q♣nq0, plus generators f✁1, e✁1, h✁1 for q♣nq1

q♣nq crystals

➓ Grantcharov, Jung, Kang, Kashiwara, Kim ‘10: Crystal bases for Uq♣q♣nqq representations (‘quantum queer supercrystals’) ➓

• ✁

➓

✁ ♣ ❜ q ✏

★ ❜ ✁ ♣ q ♣ q ✏ ♣ q ✏

✁ ♣ q ❜ ✁ ♣ ❜ q ✏

★ ❜

✁ ♣ q

♣ q ✏ ♣ q ✏

✁ ♣ q ❜

➓ ➓

q♣nq crystals

➓ Grantcharov, Jung, Kang, Kashiwara, Kim ‘10: Crystal bases for Uq♣q♣nqq representations (‘quantum queer supercrystals’) ➓ Standard queer crystal B0: 1 2 3 . . . n 1 1 ✁1 2 3 n ➓ Tensor products: Type A rules for positive arrows, and: f✁1♣b ❜ cq ✏ ★ b ❜ f✁1♣cq if wt♣bq1 ✏ wt♣bq2 ✏ 0 f✁1♣bq ❜ c

• therwise

e✁1♣b ❜ cq ✏ ★ b ❜ e✁1♣cq if wt♣bq1 ✏ wt♣bq2 ✏ 0 e✁1♣bq ❜ c

• therwise

➓ Characters: Schur P-functions ➓ QUESTION: Stembridge-like local characterization of queer crystal graphs?

q♣nq crystals

One connected component of B❜4 for q♣3q:

3 ❜ 3 ❜ 3 ❜ 1 3 ❜ 3 ❜ 3 ❜ 2 3 ❜ 2 ❜ 3 ❜ 1 3 ❜ 2 ❜ 3 ❜ 2 2 ❜ 1 ❜ 3 ❜ 1 2 ❜ 1 ❜ 2 ❜ 1 1 ❜ 3 ❜ 3 ❜ 2 1 ❜ 3 ❜ 3 ❜ 1 1 ❜ 2 ❜ 3 ❜ 2 1 ❜ 2 ❜ 3 ❜ 1 1 ❜ 2 ❜ 2 ❜ 1 1 ❜ 1 ❜ 3 ❜ 2 1 ❜ 1 ❜ 3 ❜ 1 1 ❜ 1 ❜ 2 ❜ 1 3 ❜ 2 ❜ 2 ❜ 1 3 ❜ 1 ❜ 3 ❜ 2 3 ❜ 1 ❜ 3 ❜ 1 3 ❜ 1 ❜ 2 ❜ 1 2 ❜ 3 ❜ 3 ❜ 2 2 ❜ 3 ❜ 3 ❜ 1 2 ❜ 2 ❜ 3 ❜ 2 2 ❜ 2 ❜ 3 ❜ 1 2 ❜ 2 ❜ 2 ❜ 1 2 ❜ 1 ❜ 3 ❜ 2 1 1 1 2 2 1 1 1 1 2 1 1 1 1 1 1 2 2 2 2 1 2 1 1 2 1 1 2 1 2 2 1 1 1 1 1 1 2

Notice ‘fake highest weight’ element 3 ❜ 1 ❜ 2 ❜ 1.

Restricting to ✁1, 1 or ✁1, 2 arrows

Conjecture (Assaf, Oguz ‘18)

In addition to the Stembridge axioms for the positive arrows and the assumption that ✁1 arrows commute with all i-arrows for i ➙ 3, the relations below uniquely characterize queer crystals.

✁1 ✁1 1 1 ✁1 1 1 ✁1 ✁1 1 1 1 ✁1 1 ✁1 ✁1 2 2 ✁1 2 2 ✁1 ✁1 2 2 2 ✁1 ✁1 2 2 ✁1 ✁1 2 2 2 ✁1

Restricting to ✁1, 1 or ✁1, 2 arrows

Conjecture (Assaf, Oguz ‘18)

In addition to the Stembridge axioms for the positive arrows and the assumption that ✁1 arrows commute with all i-arrows for i ➙ 3, the relations below uniquely characterize queer crystals.

✁1 ✁1 1 1 ✁1 1 1 ✁1 ✁1 1 1 1 ✁1 1 ✁1 ✁1 2 2 ✁1 2 2 ✁1 ✁1 2 2 2 ✁1 ✁1 2 2 ✁1 ✁1 2 2 2 ✁1

(GHPS) A counterexample exists!

Further axioms

Can add extra axioms to entirely characterize q♣nq crystals. Require:

Definition

Type A component graph G♣Cq: ➓ Delete ✁1 arrows; remaining arrows are ‘type A’ ➓ Replace each type A component with a single vertex labeled by highest weight; edge between them if ✁1 arrow between them.

Component graph

3 ❜ 3 ❜ 3 ❜ 1 3 ❜ 3 ❜ 3 ❜ 2 3 ❜ 2 ❜ 3 ❜ 1 3 ❜ 2 ❜ 3 ❜ 2 2 ❜ 1 ❜ 3 ❜ 1 2 ❜ 1 ❜ 2 ❜ 1 1 ❜ 3 ❜ 3 ❜ 2 1 ❜ 3 ❜ 3 ❜ 1 1 ❜ 2 ❜ 3 ❜ 2 1 ❜ 2 ❜ 3 ❜ 1 1 ❜ 2 ❜ 2 ❜ 1 1 ❜ 1 ❜ 3 ❜ 2 1 ❜ 1 ❜ 3 ❜ 1 1 ❜ 1 ❜ 2 ❜ 1 3 ❜ 2 ❜ 2 ❜ 1 3 ❜ 1 ❜ 3 ❜ 2 3 ❜ 1 ❜ 3 ❜ 1 3 ❜ 1 ❜ 2 ❜ 1 2 ❜ 3 ❜ 3 ❜ 2 2 ❜ 3 ❜ 3 ❜ 1 2 ❜ 2 ❜ 3 ❜ 2 2 ❜ 2 ❜ 3 ❜ 1 2 ❜ 2 ❜ 2 ❜ 1 2 ❜ 1 ❜ 3 ❜ 2 1 1 1 2 2 1 1 1 1 2 1 1 1 1 1 1 2 2 2 2 1 2 1 1 2 1 1 2 1 2 2 1 1 1 1 1 1 2

Component graph

3 ❜ 3 ❜ 3 ❜ 1 3 ❜ 3 ❜ 3 ❜ 2 3 ❜ 2 ❜ 3 ❜ 1 3 ❜ 2 ❜ 3 ❜ 2 2 ❜ 1 ❜ 3 ❜ 1 2 ❜ 1 ❜ 2 ❜ 1 1 ❜ 3 ❜ 3 ❜ 2 1 ❜ 3 ❜ 3 ❜ 1 1 ❜ 2 ❜ 3 ❜ 2 1 ❜ 2 ❜ 3 ❜ 1 1 ❜ 2 ❜ 2 ❜ 1 1 ❜ 1 ❜ 3 ❜ 2 1 ❜ 1 ❜ 3 ❜ 1 1 ❜ 1 ❜ 2 ❜ 1 3 ❜ 2 ❜ 2 ❜ 1 3 ❜ 1 ❜ 3 ❜ 2 3 ❜ 1 ❜ 3 ❜ 1 3 ❜ 1 ❜ 2 ❜ 1 2 ❜ 3 ❜ 3 ❜ 2 2 ❜ 3 ❜ 3 ❜ 1 2 ❜ 2 ❜ 3 ❜ 2 2 ❜ 2 ❜ 3 ❜ 1 2 ❜ 2 ❜ 2 ❜ 1 2 ❜ 1 ❜ 3 ❜ 2 1 1 1 2 2 1 1 1 2 1 1 1 2 2 2 2 2 1 2 2 2 2 1 1 1 2

Component graph

3 ❜ 3 ❜ 3 ❜ 1 3 ❜ 3 ❜ 3 ❜ 2 3 ❜ 2 ❜ 3 ❜ 1 3 ❜ 2 ❜ 3 ❜ 2 2 ❜ 1 ❜ 3 ❜ 1 2 ❜ 1 ❜ 2 ❜ 1 1 ❜ 3 ❜ 3 ❜ 2 1 ❜ 3 ❜ 3 ❜ 1 1 ❜ 2 ❜ 3 ❜ 2 1 ❜ 2 ❜ 3 ❜ 1 1 ❜ 2 ❜ 2 ❜ 1 1 ❜ 1 ❜ 3 ❜ 2 1 ❜ 1 ❜ 3 ❜ 1 1 ❜ 1 ❜ 2 ❜ 1 3 ❜ 2 ❜ 2 ❜ 1 3 ❜ 1 ❜ 3 ❜ 2 3 ❜ 1 ❜ 3 ❜ 1 3 ❜ 1 ❜ 2 ❜ 1 2 ❜ 3 ❜ 3 ❜ 2 2 ❜ 3 ❜ 3 ❜ 1 2 ❜ 2 ❜ 3 ❜ 2 2 ❜ 2 ❜ 3 ❜ 1 2 ❜ 2 ❜ 2 ❜ 1 2 ❜ 1 ❜ 3 ❜ 2 1 1 1 2 2 1 1 1 2 1 1 1 2 2 2 2 1 2 1 2 2 1 2 2 1 1 1 2

Component graph

2 ❜ 1 ❜ 2 ❜ 1 1 ❜ 1 ❜ 2 ❜ 1 3 ❜ 1 ❜ 2 ❜ 1 2 ❜ 2 ❜ 2 ❜ 1

Another component graph

4 ❜ 3 ❜ 4 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1 1 ❜ 1 ❜ 4 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1 3 ❜ 2 ❜ 4 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1 3 ❜ 2 ❜ 3 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1 3 ❜ 1 ❜ 4 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1 3 ❜ 1 ❜ 2 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1 2 ❜ 1 ❜ 4 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1 2 ❜ 1 ❜ 3 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1 2 ❜ 1 ❜ 2 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1 4 ❜ 2 ❜ 3 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1 1 ❜ 1 ❜ 3 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1 1 ❜ 1 ❜ 2 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1 4 ❜ 1 ❜ 3 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1 4 ❜ 1 ❜ 2 ❜ 1 ❜ 2 ❜ 1 ❜ 3 ❜ 2 ❜ 1

➓ Gives expansion of P-schur function Pλ in terms of Schur functions sµ.

Combinatorial description of G♣Cq

➓ Define f✁i :✏ s✁1

wi f✁1swi

and e✁i :✏ s✁1

wi e✁1swi

where wi ✏ s2 ☎ ☎ ☎ sis1 ☎ ☎ ☎ si✁1 and si is reflection along i-string ➓ Adding in ✁i arrows removes fake highest weights [GJKKK]

♣✁ q ✏ ✁

• ☎ ☎ ☎

✁

♣ q

♣✁ q

→

♣✁ q♣ q

Ñ ♣ q

✁

P

Combinatorial description of G♣Cq

➓ Define f✁i :✏ s✁1

wi f✁1swi

and e✁i :✏ s✁1

wi e✁1swi

where wi ✏ s2 ☎ ☎ ☎ sis1 ☎ ☎ ☎ si✁1 and si is reflection along i-string ➓ Adding in ✁i arrows removes fake highest weights [GJKKK] Define f♣✁i,hq :✏ f✁ifi1fi2 ☎ ☎ ☎ fh✁1.

Proposition (GHPS)

Minimal set of edges to connect G♣Cq: starting at highest weight, apply f♣✁i,hq to each vertex v for some i and h → i minimal such that f♣✁i,hq♣vq is defined. Ñ ♣ q

✁

P

Combinatorial description of G♣Cq

➓ Define f✁i :✏ s✁1

wi f✁1swi

and e✁i :✏ s✁1

wi e✁1swi

where wi ✏ s2 ☎ ☎ ☎ sis1 ☎ ☎ ☎ si✁1 and si is reflection along i-string ➓ Adding in ✁i arrows removes fake highest weights [GJKKK] Define f♣✁i,hq :✏ f✁ifi1fi2 ☎ ☎ ☎ fh✁1.

Proposition (GHPS)

Minimal set of edges to connect G♣Cq: starting at highest weight, apply f♣✁i,hq to each vertex v for some i and h → i minimal such that f♣✁i,hq♣vq is defined.

Proposition (GHPS)

Edge C1 Ñ C2 is in G♣Cq iff e✁iu2 P C1 for some i, where u2 is the highest weight element of C2.

Combinatorial description of G♣Cq (continued)

Theorem (GHPS)

There are explicit combinatorial algorithms for computing f✁i and e✁i on type A highest weight words. Algorithm for f✁i: ➓ b: highest weight word. Ex: b ✏ 545423321211 ➓ ✁ ✏ ➓ ✁ ✏ ➓

✁ ♣ q

➔ ✁

• ✁ ♣ q ✏

✁ ✁ ✁

Combinatorial description of G♣Cq (continued)

Theorem (GHPS)

There are explicit combinatorial algorithms for computing f✁i and e✁i on type A highest weight words. Algorithm for f✁i: ➓ b: highest weight word. Ex: b ✏ 545423321211 ➓ Find leftmost i, i ✁ 1, . . . , 1. Ex: b ✏ 545423321211 ➓ ✁ ✏ ➓

✁ ♣ q

➔ ✁

• ✁ ♣ q ✏

✁ ✁ ✁

Combinatorial description of G♣Cq (continued)

Theorem (GHPS)

There are explicit combinatorial algorithms for computing f✁i and e✁i on type A highest weight words. Algorithm for f✁i: ➓ b: highest weight word. Ex: b ✏ 545423321211 ➓ Find leftmost i, i ✁ 1, . . . , 1. Ex: b ✏ 545423321211 ➓ Find rightmost i, i ✁ 1, . . . , 1 from right ending at the 1. Ex: b ✏ 545423321211 ➓

✁ ♣ q

➔ ✁

• ✁ ♣ q ✏

✁ ✁ ✁

Combinatorial description of G♣Cq (continued)

Theorem (GHPS)

There are explicit combinatorial algorithms for computing f✁i and e✁i on type A highest weight words. Algorithm for f✁i: ➓ b: highest weight word. Ex: b ✏ 545423321211 ➓ Find leftmost i, i ✁ 1, . . . , 1. Ex: b ✏ 545423321211 ➓ Find rightmost i, i ✁ 1, . . . , 1 from right ending at the 1. Ex: b ✏ 545423321211 ➓ f✁i♣bq: If j ➔ j, lower j to j ✁ 1 and raise j to j 1. Ex: f✁5♣bq ✏ 436522421211.

✁ ✁ ✁

Combinatorial description of G♣Cq (continued)

Theorem (GHPS)

There are explicit combinatorial algorithms for computing f✁i and e✁i on type A highest weight words. Algorithm for f✁i: ➓ b: highest weight word. Ex: b ✏ 545423321211 ➓ Find leftmost i, i ✁ 1, . . . , 1. Ex: b ✏ 545423321211 ➓ Find rightmost i, i ✁ 1, . . . , 1 from right ending at the 1. Ex: b ✏ 545423321211 ➓ f✁i♣bq: If j ➔ j, lower j to j ✁ 1 and raise j to j 1. Ex: f✁5♣bq ✏ 436522421211. Similar algorithms for e✁i and determining if f✁i, e✁i defined.

Main Theorem: Characterization

Theorem (GHPS)

Let C be a connected component of a generic abstract queer crystal such that:

• 1. C satisfies the local axioms of Stembridge, Assaf and Oguz
• 2. The component graph G♣Cq matches G♣Dq for some

connected component D of B❜ℓ

• 3. C satisfies three extra connectivity axioms. (Put back all ✁1

arrows.) Then C is a queer supercrystal and C ✕ D.

Main Theorem: Characterization

Theorem (GHPS)

Let C be a connected component of a generic abstract queer crystal such that:

• 1. C satisfies the local axioms of Stembridge, Assaf and Oguz
• 2. The component graph G♣Cq matches G♣Dq for some

connected component D of B❜ℓ

• 3. C satisfies three extra connectivity axioms. (Put back all ✁1

arrows.) Then C is a queer supercrystal and C ✕ D.

Thank you!

Connectivity axioms: Almost lowest weight elements

✁1 ✁1 1 1 ✁1 1 1 ✁1 ✁1 1 1 1 ✁1 1 ✁1 ✁1 2 2 ✁1 2 2 ✁1 ✁1 2 2 2 ✁1 ✁1 2 2 ✁1 ✁1 2 2 2 ✁1

Almost lowest weight elements: ϕ1♣bq ✏ 2 and ϕi♣bq ✏ 0 for all i P I0③t1✉ ✏ ♣ ☎ ☎ ☎ q♣ ☎ ☎ ☎ q ↕ ↕ ↕

Connectivity axioms: Almost lowest weight elements

✁1 ✁1 1 1 ✁1 1 1 ✁1 ✁1 1 1 1 ✁1 1 ✁1 ✁1 2 2 ✁1 2 2 ✁1 ✁1 2 2 2 ✁1 ✁1 2 2 ✁1 ✁1 2 2 2 ✁1

Almost lowest weight elements: ϕ1♣bq ✏ 2 and ϕi♣bq ✏ 0 for all i P I0③t1✉

Lemma

Almost lowest weight elements are gj,k :✏ ♣e1 ☎ ☎ ☎ ejq♣e1 ☎ ☎ ☎ ekqv, where v is lowest weight and 1 ↕ j ↕ k ↕ n.

Connectivity axioms

• C0. ϕ✁1♣gj,kq ✏ 0 implies that ϕ✁1♣e1 ☎ ☎ ☎ ekvq ✏ 0.
• C1. If G♣Cq contains edge u Ñ u✶ such that wt♣u✶q is obtained

from wt♣uq by moving a box from row n 1 ✁ k to row n 1 ✁ h with h ➔ k. Then for all h ➔ j ↕ k, f✁1gj,k ✏ ♣e2 ☎ ☎ ☎ ejq♣e1 ☎ ☎ ☎ ehqv✶ where v✶ is I0-lowest weight with Ò v✶ ✏ u✶. C2.

(a) G♣Cq contains edge u Ñ u✶ such that wt♣u✶q is obtained from wt♣uq by moving a box from row n 1 ✁ k to row n 1 ✁ h with h ➔ k or (b) no such edge exists in G♣Cq

Then for all 1 ↕ j ↕ h in case (a) and all 1 ↕ j ↕ k in case (b) f✁1gj,k ✏ ♣e2 ☎ ☎ ☎ ekq♣e1 ☎ ☎ ☎ ejqv.