K-Knuth Equivalence for Increasing Tableaux

TL;DR

This paper explores K-Knuth equivalence relations on words and increasing tableaux, introducing new families of unique rectification targets, bounds on word transformations, and algorithms for classifying equivalence classes.

Contribution

It provides new families of URTs, bounds on transformation lengths, and algorithms for classifying K-Knuth equivalence classes of tableaux.

Findings

New families of URTs identified

Bound established on length of intermediate words

Algorithm developed for equivalence classification

Abstract

A K-theoretic analogue of RSK insertion and Knuth equivalence relations was first introduced in 2006 by Buch, Kresch, Shimozono, Tamvakis, and Yong. The resulting K-Knuth equivalence relations on words and increasing tableaux on [n] has prompted investigation into the equivalence classes of tableaux arising from these relations. Of particular interest are the tableaux that are unique in their class, which we refer to as unique rectification targets (URTs). In this paper we give several new families of URTs and a bound on the length of intermediate words connecting two K-Knuth equivalent words. In addition, we describe an algorithm to determine if two words are K-Knuth equivalent and to compute all K-Knuth equivalence classes of tableaux on [n].