Bumping operators and insertion algorithms for queer supercrystals

TL;DR

This paper proves that certain crystals related to involution words for self-inverse permutations are normal, identifies their components, and introduces shifted insertion algorithms as analogues of Edelman-Greene correspondence.

Contribution

It establishes the normality and component structure of Hiroshima's queer supercrystals and introduces shifted insertion algorithms to analyze their properties.

Findings

Crystals are shown to be normal and their connected components identified.

Insertion algorithms serve as shifted analogues of Edelman-Greene correspondence.

Passing to the recording tableau defines a crystal morphism.

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.