Characterization of Queer Super Crystals

Crystals of type An 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

preprint arXiv:1809.04647

Workshop on “Representation Theory, Combinatorics, and Geometry”, University of Virginia, October 19, 2018 I A superalgebra is a Z2-graded algebra G0 ⊕ G1. I A Lie superalgebra comes with a bracket operation satisfying “super” antisymmetry and the “super” Jacobi identity. I These are identical to the usual ones up to a power of −1. Setting G1 = 0 recovers the deﬁnition of Lie algebra. I In physics: uniﬁcation of bosons and fermions I In mathematics: projective representations of the symmetric group Queer super Lie algebra

I The Lie superalgebra q(n) = sl(n) ⊕ sl(n) is the natural analog to the Lie algebra An−1 = sl(n). I Highest weight crystals for queer super Lie algebras (Grantcharov et al.)

Goal Characterization of queer super crystals

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Goal

Lie superalgebras: I A Lie superalgebra comes with a bracket operation satisfying “super” antisymmetry and the “super” Jacobi identity. I These are identical to the usual ones up to a power of −1. Setting G1 = 0 recovers the deﬁnition of Lie algebra. I In physics: uniﬁcation of bosons and fermions I In mathematics: projective representations of the symmetric group Queer super Lie algebra

I The Lie superalgebra q(n) = sl(n) ⊕ sl(n) is the natural analog to the Lie algebra An−1 = sl(n). I Highest weight crystals for queer super Lie algebras (Grantcharov et al.)

Goal Characterization of queer super crystals

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Goal

Lie superalgebras: I A superalgebra is a Z2-graded algebra G0 ⊕ G1. I These are identical to the usual ones up to a power of −1. Setting G1 = 0 recovers the deﬁnition of Lie algebra. I In physics: uniﬁcation of bosons and fermions I In mathematics: projective representations of the symmetric group Queer super Lie algebra

I The Lie superalgebra q(n) = sl(n) ⊕ sl(n) is the natural analog to the Lie algebra An−1 = sl(n). I Highest weight crystals for queer super Lie algebras (Grantcharov et al.)

Goal Characterization of queer super crystals

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Goal

Lie superalgebras: I A superalgebra is a Z2-graded algebra G0 ⊕ G1. I A Lie superalgebra comes with a bracket operation satisfying “super” antisymmetry and the “super” Jacobi identity. I In physics: uniﬁcation of bosons and fermions I In mathematics: projective representations of the symmetric group Queer super Lie algebra

I The Lie superalgebra q(n) = sl(n) ⊕ sl(n) is the natural analog to the Lie algebra An−1 = sl(n). I Highest weight crystals for queer super Lie algebras (Grantcharov et al.)

Goal Characterization of queer super crystals

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Goal

I The Lie superalgebra q(n) = sl(n) ⊕ sl(n) is the natural analog to the Lie algebra An−1 = sl(n). I Highest weight crystals for queer super Lie algebras (Grantcharov et al.)

Goal Characterization of queer super crystals

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Goal

Lie superalgebras: I A superalgebra is a Z2-graded algebra G0 ⊕ G1. I A Lie superalgebra comes with a bracket operation satisfying “super” antisymmetry and the “super” Jacobi identity. I These are identical to the usual ones up to a power of −1. Setting G1 = 0 recovers the deﬁnition of Lie algebra. I In physics: uniﬁcation of bosons and fermions I In mathematics: projective representations of the symmetric group I The Lie superalgebra q(n) = sl(n) ⊕ sl(n) is the natural analog to the Lie algebra An−1 = sl(n). I Highest weight crystals for queer super Lie algebras (Grantcharov et al.)

Goal Characterization of queer super crystals

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Goal

Lie superalgebras: I A superalgebra is a Z2-graded algebra G0 ⊕ G1. I A Lie superalgebra comes with a bracket operation satisfying “super” antisymmetry and the “super” Jacobi identity. I These are identical to the usual ones up to a power of −1. Setting G1 = 0 recovers the deﬁnition of Lie algebra. I In physics: uniﬁcation of bosons and fermions I In mathematics: projective representations of the symmetric group Queer super Lie algebra I Highest weight crystals for queer super Lie algebras (Grantcharov et al.)

Goal Characterization of queer super crystals

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Goal

Lie superalgebras: I A superalgebra is a Z2-graded algebra G0 ⊕ G1. I A Lie superalgebra comes with a bracket operation satisfying “super” antisymmetry and the “super” Jacobi identity. I These are identical to the usual ones up to a power of −1. Setting G1 = 0 recovers the deﬁnition of Lie algebra. I In physics: uniﬁcation of bosons and fermions I In mathematics: projective representations of the symmetric group Queer super Lie algebra

I The Lie superalgebra q(n) = sl(n) ⊕ sl(n) is the natural analog to the Lie algebra An−1 = sl(n). Goal Characterization of queer super crystals

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Goal

Lie superalgebras: I A superalgebra is a Z2-graded algebra G0 ⊕ G1. I A Lie superalgebra comes with a bracket operation satisfying “super” antisymmetry and the “super” Jacobi identity. I These are identical to the usual ones up to a power of −1. Setting G1 = 0 recovers the deﬁnition of Lie algebra. I In physics: uniﬁcation of bosons and fermions I In mathematics: projective representations of the symmetric group Queer super Lie algebra

I The Lie superalgebra q(n) = sl(n) ⊕ sl(n) is the natural analog to the Lie algebra An−1 = sl(n). I Highest weight crystals for queer super Lie algebras (Grantcharov et al.) Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Goal

Lie superalgebras: I A superalgebra is a Z2-graded algebra G0 ⊕ G1. I A Lie superalgebra comes with a bracket operation satisfying “super” antisymmetry and the “super” Jacobi identity. I These are identical to the usual ones up to a power of −1. Setting G1 = 0 recovers the deﬁnition of Lie algebra. I In physics: uniﬁcation of bosons and fermions I In mathematics: projective representations of the symmetric group Queer super Lie algebra

I The Lie superalgebra q(n) = sl(n) ⊕ sl(n) is the natural analog to the Lie algebra An−1 = sl(n). I Highest weight crystals for queer super Lie algebras (Grantcharov et al.)

Goal Characterization of queer super crystals Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Outline

1 Crystals of type An

2 Queer supercrystals

3 Stembridge axioms

4 Characterization of queer crystals weight latticeΛ= n+1 Z>0 index set I = {1, 2,..., n} n+1 simple root αi = i − i+1, i i-th standard basis vector of Z string lengths for b ∈ B

k ϕi (b) = max{k ∈ Z>0 | fi (b) 6= 0} k εi (b) = max{k ∈ Z>0 | ei (b) 6= 0} We require: 0 0 A1. fi b = b if and only if b = ei b 0 wt(b ) = wt(b) + αi

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Crystals of type An

Abstract crystal of type An: nonempty set B together with the maps

ei , fi : B → B t {0} (i ∈ I ) wt: B → Λ string lengths for b ∈ B

k ϕi (b) = max{k ∈ Z>0 | fi (b) 6= 0} k εi (b) = max{k ∈ Z>0 | ei (b) 6= 0} We require: 0 0 A1. fi b = b if and only if b = ei b 0 wt(b ) = wt(b) + αi

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Crystals of type An

Abstract crystal of type An: nonempty set B together with the maps

ei , fi : B → B t {0} (i ∈ I ) wt: B → Λ

weight latticeΛ= n+1 Z>0 index set I = {1, 2,..., n} n+1 simple root αi = i − i+1, i i-th standard basis vector of Z We require: 0 0 A1. fi b = b if and only if b = ei b 0 wt(b ) = wt(b) + αi

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Crystals of type An

Abstract crystal of type An: nonempty set B together with the maps

ei , fi : B → B t {0} (i ∈ I ) wt: B → Λ

weight latticeΛ= n+1 Z>0 index set I = {1, 2,..., n} n+1 simple root αi = i − i+1, i i-th standard basis vector of Z string lengths for b ∈ B

k ϕi (b) = max{k ∈ Z>0 | fi (b) 6= 0} k εi (b) = max{k ∈ Z>0 | ei (b) 6= 0} Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Crystals of type An

Abstract crystal of type An: nonempty set B together with the maps

ei , fi : B → B t {0} (i ∈ I ) wt: B → Λ

weight latticeΛ= n+1 Z>0 index set I = {1, 2,..., n} n+1 simple root αi = i − i+1, i i-th standard basis vector of Z string lengths for b ∈ B

k ϕi (b) = max{k ∈ Z>0 | fi (b) 6= 0} k εi (b) = max{k ∈ Z>0 | ei (b) 6= 0} We require: 0 0 A1. fi b = b if and only if b = ei b 0 wt(b ) = wt(b) + αi wt i = i Highest weight element: 1

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Crystal: An example

Example

Standard crystal B for type An:

1 2 3 n 1 2 3 ... n + 1 Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Crystal: An example

Example

Standard crystal B for type An:

1 2 3 n 1 2 3 ... n + 1

wt i = i Highest weight element: 1 Elements: b ⊗ c := (b, c) ∈ B × C Weight map: wt(b ⊗ c) = wt(b) + wt(c) Crystal operators: ( fi (b) ⊗ c if εi (b) > ϕi (c) fi (b ⊗ c) = b ⊗ fi (c) if εi (b) < ϕi (c) ( ei (b) ⊗ c if εi (b) > ϕi (c) ei (b ⊗ c) = b ⊗ ei (c) if εi (b) 6 ϕi (c)

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Tensor products

B and C crystals of type An Deﬁnition Tensor product B ⊗ C has the following data: Weight map: wt(b ⊗ c) = wt(b) + wt(c) Crystal operators: ( fi (b) ⊗ c if εi (b) > ϕi (c) fi (b ⊗ c) = b ⊗ fi (c) if εi (b) < ϕi (c) ( ei (b) ⊗ c if εi (b) > ϕi (c) ei (b ⊗ c) = b ⊗ ei (c) if εi (b) 6 ϕi (c)

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Tensor products

B and C crystals of type An Deﬁnition Tensor product B ⊗ C has the following data: Elements: b ⊗ c := (b, c) ∈ B × C Crystal operators: ( fi (b) ⊗ c if εi (b) > ϕi (c) fi (b ⊗ c) = b ⊗ fi (c) if εi (b) < ϕi (c) ( ei (b) ⊗ c if εi (b) > ϕi (c) ei (b ⊗ c) = b ⊗ ei (c) if εi (b) 6 ϕi (c)

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Tensor products

B and C crystals of type An 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 An Queer supercrystals Stembridge axioms Characterization of queer crystals Tensor products

B and C crystals of type An 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: ( fi (b) ⊗ c if εi (b) > ϕi (c) fi (b ⊗ c) = b ⊗ fi (c) if εi (b) < ϕi (c) ( ei (b) ⊗ c if εi (b) > ϕi (c) ei (b ⊗ c) = b ⊗ ei (c) if εi (b) 6 ϕi (c) Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Example: Tensor product

Example ⊗3 Components of crystal of words B = B ⊗ B ⊗ B of type A2: 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

3 ⊗ 3 ⊗ 3 Characters: character of highest weight crystal B(λ) is Schur function sλ Littlewood–Richardson rule:

X ν sλsµ = cλµsν ν

ν cλµ = number of highest weights of weight ν in B(λ) ⊗ B(µ)

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Motivation

Why are crystals interesting? Littlewood–Richardson rule:

X ν sλsµ = cλµsν ν

ν cλµ = number of highest weights of weight ν in B(λ) ⊗ B(µ)

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Motivation

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

Why are crystals interesting? Characters: character of highest weight crystal B(λ) is Schur function sλ Littlewood–Richardson rule:

X ν sλsµ = cλµsν ν

ν cλµ = number of highest weights of weight ν in B(λ) ⊗ B(µ) Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Outline

1 Crystals of type An

2 Queer supercrystals

3 Stembridge axioms

4 Characterization of queer crystals [Grantcharov, Jung, Kang, Kashiwara, Kim, ’10]: Crystal basis theory for queer Lie superalgebras using Uq(q(n))

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Queer crystal: Developments

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

Queer Lie superalgebra q(n): a super analogue of sl(n) [Grantcharov, Jung, Kang, Kashiwara, Kim, ’10]: Crystal basis theory for queer Lie superalgebras using Uq(q(n))

I Introduced queer crystals on words with tensor product rule. I Explicit combinatorial realization of queer crystals using semistandard decomposition tableaux. I Existence of fake highest (and lowest) weights on queer crystals. 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 An 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 An 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 An Queer supercrystals Stembridge axioms Characterization of queer crystals Queer crystal: Example

One connected component of B⊗4 for q(3):

1 ⊗ 1 ⊗ 2 ⊗ 1

1 2 1 2 ⊗ 1 ⊗ 2 ⊗ 1 1 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 2 ⊗ 2 ⊗ 1

2 1 1 2 1 1 3 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 1 ⊗ 3 ⊗ 2 1 ⊗ 2 ⊗ 3 ⊗ 1 2 ⊗ 2 ⊗ 2 ⊗ 1

1 1 2 1 1 1 2 1 1 2 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 1 2 1 1 2 1 1 2 3 ⊗ 2 ⊗ 3 ⊗ 1 3 ⊗ 1 ⊗ 3 ⊗ 2 2 ⊗ 2 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 3 ⊗ 1

1 1 2 1 1 2 3 ⊗ 2 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 3 ⊗ 2 3 ⊗ 3 ⊗ 3 ⊗ 1

2 1 1 3 ⊗ 3 ⊗ 3 ⊗ 2 Littlewood–Richardson rule:

X ν PλPµ = gλµPν ν

ν gλµ = number of highest weights of weight ν in B(λ) ⊗ B(µ)

Crystals of type An 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 Pλ Crystals of type An 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 Pλ Littlewood–Richardson rule:

X ν PλPµ = gλµPν ν

ν gλµ = number of highest weights of weight ν in B(λ) ⊗ B(µ) Deﬁnition

f−i := s −1 f−1swi and e−i := s −1 e−1swi wi wi

where wi = s2 ··· si s1 ··· si−1 and si is the reﬂection along the i-string

Theorem (Grantcharov et al. 2014) Each connected component in B⊗` has a unique highest weight element with ei u = 0 and e−i u = 0 for all i ∈ I0 = {1, 2,..., n} Similarly

0 0 f−i := sw0 e−(n+1−i)sw0 and e−i := sw0 f−(n+1−i)sw0

where w0 is long word in Sn+1, give lowest weight elements.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals (Fake) highest weight elements

In the queer crystal there exist fake highest weight elements. Question: How do we detect highest weight elements? Theorem (Grantcharov et al. 2014) Each connected component in B⊗` has a unique highest weight element with ei u = 0 and e−i u = 0 for all i ∈ I0 = {1, 2,..., n} Similarly

0 0 f−i := sw0 e−(n+1−i)sw0 and e−i := sw0 f−(n+1−i)sw0

where w0 is long word in Sn+1, give lowest weight elements.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals (Fake) highest weight elements

In the queer crystal there exist fake highest weight elements. Question: How do we detect highest weight elements? Deﬁnition

f−i := s −1 f−1swi and e−i := s −1 e−1swi wi wi

where wi = s2 ··· si s1 ··· si−1 and si is the reﬂection along the i-string Similarly

0 0 f−i := sw0 e−(n+1−i)sw0 and e−i := sw0 f−(n+1−i)sw0

where w0 is long word in Sn+1, give lowest weight elements.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals (Fake) highest weight elements

In the queer crystal there exist fake highest weight elements. Question: How do we detect highest weight elements? Deﬁnition

f−i := s −1 f−1swi and e−i := s −1 e−1swi wi wi

where wi = s2 ··· si s1 ··· si−1 and si is the reﬂection along the i-string

Theorem (Grantcharov et al. 2014) Each connected component in B⊗` has a unique highest weight element with ei u = 0 and e−i u = 0 for all i ∈ I0 = {1, 2,..., n} Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals (Fake) highest weight elements

In the queer crystal there exist fake highest weight elements. Question: How do we detect highest weight elements? Deﬁnition

f−i := s −1 f−1swi and e−i := s −1 e−1swi wi wi

where wi = s2 ··· si s1 ··· si−1 and si is the reﬂection along the i-string

Theorem (Grantcharov et al. 2014) Each connected component in B⊗` has a unique highest weight element with ei u = 0 and e−i u = 0 for all i ∈ I0 = {1, 2,..., n} Similarly

0 0 f−i := sw0 e−(n+1−i)sw0 and e−i := sw0 f−(n+1−i)sw0

where w0 is long word in Sn+1, give lowest weight elements. Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Queer crystal: Example revisited

Same connected component of B⊗4:

1 ⊗ 1 ⊗ 2 ⊗ 1

1 20 2 2 10 1

2 ⊗ 1 ⊗ 2 ⊗ 1 1 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 2 ⊗ 2 ⊗ 1

2 2 20 1 1 10 10 20 1 1 2 2

3 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 1 ⊗ 3 ⊗ 2 2 ⊗ 2 ⊗ 2 ⊗ 1 1 ⊗ 2 ⊗ 3 ⊗ 1

1 1 10 20 2 2 10 1 1 1 20 2 2 20 2 2 1 10 1

3 ⊗ 2 ⊗ 2 ⊗ 1 3 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 2 ⊗ 3 ⊗ 2 2 ⊗ 1 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 3 ⊗ 1 2 ⊗ 2 ⊗ 3 ⊗ 1

2 20 2 1 1 10 2 1 1 2 2 10 20 10 1 1 20 2 2

3 ⊗ 2 ⊗ 3 ⊗ 1 3 ⊗ 1 ⊗ 3 ⊗ 2 2 ⊗ 2 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 3 ⊗ 1

10 1 1 2 2 20 1 1 20 2 10 2

3 ⊗ 2 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 3 ⊗ 2 3 ⊗ 3 ⊗ 3 ⊗ 1

20 2 2 10 1 1

3 ⊗ 3 ⊗ 3 ⊗ 2 Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Outline

1 Crystals of type An

2 Queer supercrystals

3 Stembridge axioms

4 Characterization of queer crystals [Stembridge ’03] Yes, for crystals of simply-laced root systems (in particular type An) Local rules characterize Stembridge crystals: allows pure combinatorial analysis of these crystals

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Stembridge axioms

Question Is there a local characterization of a crystal graph? Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Stembridge axioms

Question Is there a local characterization of a crystal graph?

[Stembridge ’03] Yes, for crystals of simply-laced root systems (in particular type An) Local rules characterize Stembridge crystals: allows pure combinatorial analysis of these crystals Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Stembridge axioms

B crystal for a simply-laced root system with index set I = {1, 2 ..., n}.

Axiom S1. For distinct i, j ∈ I and x, y ∈ B with y = ei x, then either εj (y) = εj (x) + 1 or εj (y) = εj (x).

• •

• • •

ei x • • • •

ei x • • • • x • • • • x • • •

• • Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Stembridge axioms

Axiom S2. For distinct i, j ∈ I , if x ∈ B with both εi (x) > 0 and εj (x) = εj (ei x) > 0, then ei ej x = ej ei x and ϕi (ej x) = ϕi (x).

0 Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Stembridge axioms

Axiom S3. For distinct i, j ∈ I , if x ∈ B with both εj (ei x) = εj (x) + 1 > 1 and εi (ej x) = εi (x) + 1 > 1, then 2 2 2 ei ej ei x = ej ei ej x 6= 0, ϕi (ej x) = ϕi (ej ei x) and 2 ϕj (ei x) = ϕj (ei ej x).

1 1 Theorem (Stembridge 2003) Every connected component of a Stembridge crystal has a unique highest weight element.

Theorem (Stembridge 2003) B, B0 Stembridge crystals, u ∈ B, u0 ∈ B0 highest weight elements If wt(u) = wt(u0), then B and B0 are isomorphic.

Stembridge crystals describe the representation theory of the corresponding Lie algebra.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Why Stembridge crystals?

Theorem (Stembridge 2003) B, C Stembridge crystals =⇒ B ⊗ C Stembridge crystal Theorem (Stembridge 2003) B, B0 Stembridge crystals, u ∈ B, u0 ∈ B0 highest weight elements If wt(u) = wt(u0), then B and B0 are isomorphic.

Stembridge crystals describe the representation theory of the corresponding Lie algebra.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Why Stembridge crystals?

Theorem (Stembridge 2003) B, C Stembridge crystals =⇒ B ⊗ C Stembridge crystal

Theorem (Stembridge 2003) Every connected component of a Stembridge crystal has a unique highest weight element. Stembridge crystals describe the representation theory of the corresponding Lie algebra.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Why Stembridge crystals?

Theorem (Stembridge 2003) B, C Stembridge crystals =⇒ B ⊗ C Stembridge crystal

Theorem (Stembridge 2003) Every connected component of a Stembridge crystal has a unique highest weight element.

Theorem (Stembridge 2003) B, B0 Stembridge crystals, u ∈ B, u0 ∈ B0 highest weight elements If wt(u) = wt(u0), then B and B0 are isomorphic. Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Why Stembridge crystals?

Theorem (Stembridge 2003) B, C Stembridge crystals =⇒ B ⊗ C Stembridge crystal

Theorem (Stembridge 2003) Every connected component of a Stembridge crystal has a unique highest weight element.

Theorem (Stembridge 2003) B, B0 Stembridge crystals, u ∈ B, u0 ∈ B0 highest weight elements If wt(u) = wt(u0), then B and B0 are isomorphic.

Stembridge crystals describe the representation theory of the corresponding Lie algebra. Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Outline

1 Crystals of type An

2 Queer supercrystals

3 Stembridge axioms

4 Characterization of queer crystals Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Stembridge type axioms

Conjecture (Assaf, Oguz 2018) In addition to the Stembridge axioms, the relations below uniquely characterize queer crystals.

−1 −1

1 1 2 2 −1 −1 −1

1 1 2 2 2 2 −1 −1 −1

−1 −1 −1

1 1 2 2 2 2 −1 −1 −1

1 2 2 −1

−1 1 Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Subcrystal example

For instance, the {−1, 2}-subcrystal of

1 ⊗ 1 ⊗ 2 ⊗ 1

1 2 1 2 ⊗ 1 ⊗ 2 ⊗ 1 1 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 2 ⊗ 2 ⊗ 1

2 1 1 2 1 1 3 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 1 ⊗ 3 ⊗ 2 1 ⊗ 2 ⊗ 3 ⊗ 1 2 ⊗ 2 ⊗ 2 ⊗ 1

1 1 2 1 1 1 2 1 1 2 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 1 2 1 1 2 1 1 2 3 ⊗ 2 ⊗ 3 ⊗ 1 3 ⊗ 1 ⊗ 3 ⊗ 2 2 ⊗ 2 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 3 ⊗ 1

1 1 2 1 1 2 3 ⊗ 2 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 3 ⊗ 2 3 ⊗ 3 ⊗ 3 ⊗ 1

2 1 1 3 ⊗ 3 ⊗ 3 ⊗ 2 Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Subcrystal Example Cont.

...is given by

1 ⊗ 1 ⊗ 2 ⊗ 1

1 2 2 ⊗ 1 ⊗ 2 ⊗ 1 1 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 2 ⊗ 2 ⊗ 1

2 1 2 1 3 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 1 ⊗ 3 ⊗ 2 1 ⊗ 2 ⊗ 3 ⊗ 1 2 ⊗ 2 ⊗ 2 ⊗ 1

1 2 1 2 1 2 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 3 ⊗ 2 ⊗ 3 ⊗ 1 3 ⊗ 1 ⊗ 3 ⊗ 2 2 ⊗ 2 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 3 ⊗ 1

1 2 1 2 3 ⊗ 2 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 3 ⊗ 2 3 ⊗ 3 ⊗ 3 ⊗ 1

2 1 3 ⊗ 3 ⊗ 3 ⊗ 2 Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Counterexample

[Gillespie, Hawkes, Poh, S. 2018]

1 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1

−1 1 2

1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 2 ⊗ 1 1 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 3 ⊗ 1 2 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1 1 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1

−1 1 2 1 2 2 1

2 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 2 ⊗ 1 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 3 ⊗ 1 1 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 2 ⊗ 1 2 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1

2 1 1 2 2 1 2 1 1 2 1 2 2 1

2 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 3 ⊗ 1 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 3 ⊗ 2 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 3 ⊗ 1 1 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 1 2 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 3 ⊗ 2 ⊗ 2 ⊗ 2 ⊗ 1 1 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 3 ⊗ 1 3 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 2 ⊗ 1

−1 2 1 2 1 2 1 1 2 2 1 2 2 1 1 1 2

2 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 3 ⊗ 1 2 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 3 ⊗ 2 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 1 ⊗ 3 ⊗ 3 ⊗ 1 1 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 1 2 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 3 ⊗ 2 3 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 3 ⊗ 1 1 ⊗ 3 ⊗ 2 ⊗ 2 ⊗ 3 ⊗ 1 2 ⊗ 3 ⊗ 2 ⊗ 2 ⊗ 2 ⊗ 1 3 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 2 ⊗ 1 3 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 3 ⊗ 1

−1 −1 2 2 1 2 1 1 1 2 1 2 1 2 1 2 1 2

2 ⊗ 3 ⊗ 1 ⊗ 3 ⊗ 3 ⊗ 1 2 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 1 ⊗ 3 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 2 ⊗ 2 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 3 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 1 2 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 3 ⊗ 2 3 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 2 ⊗ 3 ⊗ 3 ⊗ 1 2 ⊗ 3 ⊗ 2 ⊗ 2 ⊗ 3 ⊗ 1 3 ⊗ 3 ⊗ 2 ⊗ 2 ⊗ 2 ⊗ 1 3 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 3 ⊗ 1

−1 −1 2 1 2 1 2 1 2 1 2 1 1 2 2 1

3 ⊗ 3 ⊗ 1 ⊗ 3 ⊗ 3 ⊗ 1 2 ⊗ 3 ⊗ 1 ⊗ 3 ⊗ 3 ⊗ 2 1 ⊗ 3 ⊗ 2 ⊗ 3 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 2 ⊗ 2 ⊗ 3 ⊗ 2 3 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 3 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 2 ⊗ 3 ⊗ 3 ⊗ 1 3 ⊗ 3 ⊗ 2 ⊗ 2 ⊗ 3 ⊗ 1

−1 1 2 1 2 1 2

3 ⊗ 3 ⊗ 1 ⊗ 3 ⊗ 3 ⊗ 2 2 ⊗ 3 ⊗ 2 ⊗ 3 ⊗ 3 ⊗ 2 3 ⊗ 3 ⊗ 2 ⊗ 2 ⊗ 3 ⊗ 2 3 ⊗ 3 ⊗ 2 ⊗ 3 ⊗ 3 ⊗ 1

1 2

3 ⊗ 3 ⊗ 2 ⊗ 3 ⊗ 3 ⊗ 2 Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Main theorem: characterization of queer supercrystals

Theorem (GHPS 2018) C connected component of a generic abstract queer crystal satisfying: 1 C satisfies the local queer axioms. 2 C satisfies the connectivity axioms. 3 Component graph G(C) =∼ G(D) D some connected component of B⊗` Then the queer supercrystals C =∼ D. Definition

Edge from vertex C1 to vertex C2, if ∃ b1 ∈ C1 and b2 ∈ C2 such that

f−1b1 = b2.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Graph on type A components

Question: How are the type A components glued together? Vertices of G(C) are the type A components of C, labeled by highest weight elements

Edge from vertex C1 to vertex C2, if ∃ b1 ∈ C1 and b2 ∈ C2 such that

f−1b1 = b2.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Graph on type A components

Question: How are the type A components glued together?

Deﬁnition

C crystal with index set I0 ∪ {−1}, An Stembridge crystal when restricted to I0 Type A component graph G(C) deﬁned as follows: Edge from vertex C1 to vertex C2, if ∃ b1 ∈ C1 and b2 ∈ C2 such that

f−1b1 = b2.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Graph on type A components

Question: How are the type A components glued together?

Deﬁnition

C crystal with index set I0 ∪ {−1}, An Stembridge crystal when restricted to I0 Type A component graph G(C) deﬁned as follows: Vertices of G(C) are the type A components of C, labeled by highest weight elements Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Graph on type A components

Question: How are the type A components glued together?

Deﬁnition

Edge from vertex C1 to vertex C2, if ∃ b1 ∈ C1 and b2 ∈ C2 such that

f−1b1 = b2. counterexample

1 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1

2 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1 1 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1

1 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1

3 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Graph on type A components: example

correct graph G(C)

1 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1

2 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1 1 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1

1 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1

3 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Graph on type A components: example

correct graph G(C) counterexample

1 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1 1 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1

2 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1 1 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 2 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1 1 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1

1 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1 1 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 2 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 1

3 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 3 ⊗ 3 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 Edges described by e−i : Proposition (GHPS 2018)

C1, C2 distinct type A components in C Let u2 ∈ C2 be I0-highest weight element

There is an edge from C1 to C2 in G(C) ⇔ e−i u2 ∈ C1 for some i ∈ I0

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Combinatorial description of G(C)

Goal Give a combinatorial description of G(C). There is an edge from C1 to C2 in G(C) ⇔ e−i u2 ∈ C1 for some i ∈ I0

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Combinatorial description of G(C)

Goal Give a combinatorial description of G(C).

Edges described by e−i : Proposition (GHPS 2018)

C1, C2 distinct type A components in C Let u2 ∈ C2 be I0-highest weight element Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Combinatorial description of G(C)

Goal Give a combinatorial description of G(C).

Edges described by e−i : Proposition (GHPS 2018)

C1, C2 distinct type A components in C Let u2 ∈ C2 be I0-highest weight element

There is an edge from C1 to C2 in G(C) ⇔ e−i u2 ∈ C1 for some i ∈ I0 Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Combinatorial description of G(C) (continued)

Remove by-pass arrows: 1 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

1 ⊗ 1 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

1 ⊗ 1 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 2 ⊗ 1 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

2 ⊗ 1 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

3 ⊗ 1 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 3 ⊗ 2 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 4 ⊗ 1 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

3 ⊗ 2 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 4 ⊗ 2 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

4 ⊗ 3 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 Theorem (GHPS 2018) Let C be a connected component in B⊗`. Then each non by-pass edge in G(C) can be obtained by f(−i,h) for some i and h > i minimal such that f(−i,h) applies.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Combinatorial description of G(C) (continued)

Combinatorial description of remaining arrows: Deﬁne f(−i,h) := f−i fi+1fi+2 ··· fh−1. Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Combinatorial description of G(C) (continued)

Combinatorial description of remaining arrows: Deﬁne f(−i,h) := f−i fi+1fi+2 ··· fh−1. Theorem (GHPS 2018) Let C be a connected component in B⊗`. Then each non by-pass edge in G(C) can be obtained by f(−i,h) for some i and h > i minimal such that f(−i,h) applies. Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Combinatorial description of G(C) (continued)

1 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

(−2, 3) (−1, 2)

1 ⊗ 1 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

(−3, 4) (−1, 2) (−2, 3)

1 ⊗ 1 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 2 ⊗ 1 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

(−1, 2) (−3, 4) (−1, 3) (−3, 4)

2 ⊗ 1 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

(−2, 3) 3 ⊗ 2 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 (−2, 3)

3 ⊗ 1 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 (−3, 4) 4 ⊗ 1 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

(−1, 2) (−1, 2)

3 ⊗ 2 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 4 ⊗ 2 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

(−2, 4)

4 ⊗ 3 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 Set r1 = q1

Recursively rj < rj−1 for 1 < j 6 i maximal such that brj = j. By deﬁnition qj 6 rj . Let 1 6 k 6 i be maximal such that qk = rk .

Example b = 1331242312111 and i = 3

We overline bqj b = 1331242312111

and underline brj b = 1331242312111 Here k = 1.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Combinatorial description of f−i

bqi , bqi−1 ,..., bq1 leftmost sequence i, i − 1,..., 1 from left to right Recursively rj < rj−1 for 1 < j 6 i maximal such that brj = j. By deﬁnition qj 6 rj . Let 1 6 k 6 i be maximal such that qk = rk .

Example b = 1331242312111 and i = 3

We overline bqj b = 1331242312111

and underline brj b = 1331242312111 Here k = 1.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Combinatorial description of f−i

bqi , bqi−1 ,..., bq1 leftmost sequence i, i − 1,..., 1 from left to right Set r1 = q1 By deﬁnition qj 6 rj . Let 1 6 k 6 i be maximal such that qk = rk .

Example b = 1331242312111 and i = 3

We overline bqj b = 1331242312111

and underline brj b = 1331242312111 Here k = 1.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Combinatorial description of f−i

bqi , bqi−1 ,..., bq1 leftmost sequence i, i − 1,..., 1 from left to right Set r1 = q1

Recursively rj < rj−1 for 1 < j 6 i maximal such that brj = j. Example b = 1331242312111 and i = 3

We overline bqj b = 1331242312111

and underline brj b = 1331242312111 Here k = 1.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Combinatorial description of f−i

bqi , bqi−1 ,..., bq1 leftmost sequence i, i − 1,..., 1 from left to right Set r1 = q1

Recursively rj < rj−1 for 1 < j 6 i maximal such that brj = j. By deﬁnition qj 6 rj . Let 1 6 k 6 i be maximal such that qk = rk . and underline brj b = 1331242312111 Here k = 1.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Combinatorial description of f−i

bqi , bqi−1 ,..., bq1 leftmost sequence i, i − 1,..., 1 from left to right Set r1 = q1

Recursively rj < rj−1 for 1 < j 6 i maximal such that brj = j. By deﬁnition qj 6 rj . Let 1 6 k 6 i be maximal such that qk = rk .

Example b = 1331242312111 and i = 3

We overline bqj b = 1331242312111 Here k = 1.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Combinatorial description of f−i

bqi , bqi−1 ,..., bq1 leftmost sequence i, i − 1,..., 1 from left to right Set r1 = q1

Recursively rj < rj−1 for 1 < j 6 i maximal such that brj = j. By deﬁnition qj 6 rj . Let 1 6 k 6 i be maximal such that qk = rk .

Example b = 1331242312111 and i = 3

We overline bqj b = 1331242312111

and underline brj b = 1331242312111 Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Combinatorial description of f−i

bqi , bqi−1 ,..., bq1 leftmost sequence i, i − 1,..., 1 from left to right Set r1 = q1

Recursively rj < rj−1 for 1 < j 6 i maximal such that brj = j. By deﬁnition qj 6 rj . Let 1 6 k 6 i be maximal such that qk = rk .

Example b = 1331242312111 and i = 3

We overline bqj b = 1331242312111

and underline brj b = 1331242312111 Here k = 1. Example

b = 1331242312111 i = 3

f−3(b) = 1241143322111

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Combinatorial description of f−i (continued)

bqi , bqi−1 ,..., bq1 leftmost sequence i, i − 1,..., 1 from left to right Set r1 = q1

Recursively rj < rj−1 for 1 < j 6 i maximal such that brj = j. By deﬁnition qj 6 rj . Let 1 6 k 6 i be maximal such that qk = rk . Proposition ⊗` Let b ∈ B be {1, 2,..., i}-highest weight for i ∈ I0 and ϕ−i (b) = 1. Then f−i (b) is obtained from b by changing

bqj = j to j − 1 for j = i, i − 1,..., k + 1

brj = j to j + 1 for j = i, i − 1,..., k. Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

Combinatorial description of f−i (continued)

bqi , bqi−1 ,..., bq1 leftmost sequence i, i − 1,..., 1 from left to right Set r1 = q1

bqj = j to j − 1 for j = i, i − 1,..., k + 1

brj = j to j + 1 for j = i, i − 1,..., k.

Example

b = 1331242312111 i = 3

f−3(b) = 1241143322111 Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

1 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

(−2, 3) (−1, 2)

1 ⊗ 1 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

(−3, 4) (−1, 2) (−2, 3)

1 ⊗ 1 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 2 ⊗ 1 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

(−1, 2) (−3, 4) (−1, 3) (−3, 4)

2 ⊗ 1 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

(−2, 3) 3 ⊗ 2 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 (−2, 3)

3 ⊗ 1 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 (−3, 4) 4 ⊗ 1 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

(−1, 2) (−1, 2)

3 ⊗ 2 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 4 ⊗ 2 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1

(−2, 4)

4 ⊗ 3 ⊗ 4 ⊗ 1 ⊗ 2 ⊗ 1 ⊗ 3 ⊗ 2 ⊗ 1 Lemma

Almost lowest weight elements are gj,k := (e1 ··· ej )(e1 ··· ek )v, where v is lowest weight and 1 6 j 6 k 6 n.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Almost lowest weight elements

−1 −1

1 1 2 2 −1 −1 −1

1 1 2 2 2 2 −1 −1 −1

−1 −1 −1

1 1 2 2 2 2 −1 −1 −1

1 2 2 −1

−1 1

Almost lowest weight elements:

ϕ1(b) = 2 and ϕi (b) = 0 for all i ∈ I0 \{1} Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Almost lowest weight elements

−1 −1

1 1 2 2 −1 −1 −1

1 1 2 2 2 2 −1 −1 −1

−1 −1 −1

1 1 2 2 2 2 −1 −1 −1

1 2 2 −1

−1 1

Almost lowest weight elements:

ϕ1(b) = 2 and ϕi (b) = 0 for all i ∈ I0 \{1}

Lemma

Almost lowest weight elements are gj,k := (e1 ··· ej )(e1 ··· ek )v, where v is lowest weight and 1 6 j 6 k 6 n. C1. If G(C) contains edge u → u0 such that wt(u0) is obtained from wt(u) by moving a box from row n + 1 − k to row n + 1 − h with h < k. Then for all h < j 6 k

0 f−1gj,k = (e2 ··· ej )(e1 ··· eh)v ,

0 0 0 where v is I0-lowest weight with ↑ v = u . C2. (a) G(C) contains edge u → u0 such that wt(u0) is obtained from wt(u) 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(C) Then for all1 6 j 6 h in case (a) and all1 6 j 6 k in case (b)

f−1gj,k = (e2 ··· ek )(e1 ··· ej )v.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Connectivity axioms

Deﬁnition (Connectivity axioms)

C0. ϕ−1(gj,k ) = 0 implies that ϕ−1(e1 ··· ek v) = 0. C2. (a) G(C) contains edge u → u0 such that wt(u0) is obtained from wt(u) 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(C) Then for all1 6 j 6 h in case (a) and all1 6 j 6 k in case (b)

f−1gj,k = (e2 ··· ek )(e1 ··· ej )v.

Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Connectivity axioms

Deﬁnition (Connectivity axioms)

C0. ϕ−1(gj,k ) = 0 implies that ϕ−1(e1 ··· ek v) = 0. C1. If G(C) contains edge u → u0 such that wt(u0) is obtained from wt(u) by moving a box from row n + 1 − k to row n + 1 − h with h < k. Then for all h < j 6 k

0 f−1gj,k = (e2 ··· ej )(e1 ··· eh)v ,

0 0 0 where v is I0-lowest weight with ↑ v = u . Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals Connectivity axioms

Deﬁnition (Connectivity axioms)

0 f−1gj,k = (e2 ··· ej )(e1 ··· eh)v ,

f−1gj,k = (e2 ··· ek )(e1 ··· ej )v. Crystals of type An Queer supercrystals Stembridge axioms Characterization of queer crystals

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