Coxeter-Knuth graphs and a signed Little map for type B reduced words

TL;DR

This paper introduces a type B analog of Little's algorithm, preserving key combinatorial structures and providing a bijective framework for type B Schubert calculus, extending many type A results.

Contribution

It develops a new type B Little map, proves its properties, and characterizes shifted dual equivalence graphs to understand Coxeter-Knuth relations.

Findings

The algorithm preserves the Kraśkiewicz insertion tableau.

Provides a bijective realization of type B transition equations.

Characterizes shifted dual equivalence graphs axiomatically.

Abstract

We define an analog of David Little's algorithm for reduced words in type B, and investigate its main properties. In particular, we show that our algorithm preserves the recording tableau of Kra\'{s}kiewicz insertion, and that it provides a bijective realization of the Type B transition equations in Schubert calculus. Many other aspects of type A theory carry over to this new setting. Our primary tool is a shifted version of the dual equivalence graphs defined by Assaf and further developed by Roberts. We provide an axiomatic characterization of shifted dual equivalence graphs, and use them to prove a structure theorem for the graph of Type B Coxeter-Knuth relations.