Queer Supercrystal Structure for Increasing Factorizations of Fixed-Point-Free Involution Words

TL;DR

This paper demonstrates that the set of increasing factorizations of fixed-point-free involution words forms a queer supercrystal structure, linking combinatorial objects via symplectic shifted Hecke insertion.

Contribution

It establishes a novel correspondence between fixed-point-free involution words and primed tableaux using symplectic shifted Hecke insertion, revealing a queer supercrystal structure.

Findings

Fixed-point-free involution words have a queer supercrystal structure.

Symplectic shifted Hecke insertion creates a bijection with primed tableaux.

Coxeter-Knuth related words share the same insertion tableau.

Abstract

We show that the set of increasing factorizations of fixed-point-free (FPF) involution words has the structure of queer supercrystals. By exploiting the algorithm of symplectic shifted Hecke insertion recently introduced by Marberg, we establish the one-to-one correspondence between the set of increasing factorizations of fixed-point-free involution words and the set of primed tableau (semistandard marked shifted tableaux) and the latter admits the structure of queer supercrystals. In order to establish the correspondence, we prove that the Coxeter-Knuth related FPF-involution words have the same insertion tableau in the symplectic shifted Hecke insertion, where the insertion tableau is an increasing shifted tableau and the recording tableau is a primed tableau.