Bumping operators and insertion algorithms for queer supercrystals

Abstract

Results of Morse and Schilling show that the set of increasing factorizations of reduced words for a permutation is naturally a crystal for the general linear Lie algebra. Hiroshima has recently constructed two superalgebra analogues of such crystals. Specifically, Hiroshima has shown that the sets of increasing factorizations of involution words and fpf-involution words for a self-inverse permutation are each crystals for the queer Lie superalgebra. In this paper, we prove that these crystals are normal and identify their connected components. To accomplish this, we study two insertion algorithms that may be viewed as shifted analogues of the Edelman-Greene correspondence. We prove that the connected components of Hiroshima's crystals are the subsets of factorizations with the same insertion tableau for these algorithms, and that passing to the recording tableau defines a crystal morphism. This confirms a conjecture of Hiroshima. Our methods involve a detailed investigation of certain analogues of the Little map, through which we extend several results of Hamaker and Young.

arXiv comment54 pages, 3 figures; v2: updated references, fixed typos, improved notation; v3: expository appendix on tableau crystal operators added along with several new examples, expanded discussion of tableau dual equivalence operators, several corrections; v4: fixed typos, added remarks in Section 3, final version

References in corpus (13)

• Involution words II: braid relations and atomic structures
• Schur P-positivity and involution Stanley symmetric functions
• Highest weight modules over quantum queer Lie superalgebra U_q(q(n))
• Polynomials for symmetric orbit closures in the flag variety
• Transition formulas for involution Schubert polynomials
• Dual equivalence graphs I: A new paradigm for Schur positivity
• Crystal approach to affine Schubert calculus
• A symplectic refinement of shifted Hecke insertion
• Characterization of queer supercrystals
• A local characterization of crystals for the quantum queer superalgebra
• $\mathfrak{q}$-crystal structure on primed tableaux and on signed unimodal factorizations of reduced words of type $B$
• On some actions of the 0-Hecke monoids of affine symmetric groups
• Quasiparabolic sets and Stanley symmetric functions for affine fixed-point-free involutions

Cited by in corpus (7)

• A symplectic refinement of shifted Hecke insertion
• Shifted insertion algorithms for primed words
• Extending a word property for twisted Coxeter systems
• Highest weight crystals for Schur Q-functions
• Primed decomposition tableaux and extended queer crystals
• Crystals for shifted key polynomials
• Crystals for set-valued decomposition tableaux