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 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. 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 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 = f1; 2;:::; ng n+1 simple root αi = i − i+1, i i-th standard basis vector of Z string lengths for b 2 B k 'i (b) = maxfk 2 Z>0 j fi (b) 6= 0g k "i (b) = maxfk 2 Z>0 j ei (b) 6= 0g 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 f0g (i 2 I ) wt: B ! Λ string lengths for b 2 B k 'i (b) = maxfk 2 Z>0 j fi (b) 6= 0g k "i (b) = maxfk 2 Z>0 j ei (b) 6= 0g 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 f0g (i 2 I ) wt: B ! Λ weight latticeΛ= n+1 Z>0 index set I = f1; 2;:::; ng 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 f0g (i 2 I ) wt: B ! Λ weight latticeΛ= n+1 Z>0 index set I = f1; 2;:::; ng n+1 simple root αi = i − i+1, i i-th standard basis vector of Z string lengths for b 2 B k 'i (b) = maxfk 2 Z>0 j fi (b) 6= 0g k "i (b) = maxfk 2 Z>0 j ei (b) 6= 0g 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 f0g (i 2 I ) wt: B ! Λ weight latticeΛ= n+1 Z>0 index set I = f1; 2;:::; ng n+1 simple root αi = i − i+1, i i-th standard basis vector of Z string lengths for b 2 B k 'i (b) = maxfk 2 Z>0 j fi (b) 6= 0g k "i (b) = maxfk 2 Z>0 j ei (b) 6= 0g We require: 0 0 A1.