K-Knuth Equivalence for Increasing Tableaux

Abstract

A K-theoretic analogue of RSK insertion and the Knuth equivalence relations were introduced in [1, 2]. The resulting K-Knuth equivalence relations on words and increasing tableaux on [n] have 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].

Linear Compactness and Combinatorial Bialgebras

“…There are finitely many increasing tableaux with all letters in a given finite set [37,Lemma 3.2] and every K-Knuth equivalence class contains at least one increasing tableau [13,Lemma 58]. Thus, K-Knuth equivalence is of finite-type and a somewhat improved way of indexing the elements of K (∼) n is to define [[T, n]] = w∼T [w, n] for each increasing tableau T with max(T )…”

“…There is an algorithm to compute all K-Knuth classes of words with a given set of letters, however [13].…”

Hecke insertion and maximal increasing and decreasing sequences in fillings of stack polyominoes

“…For the convenience of the reader, in this section we give all the necessary definitions, following the original description and notation as much as possible. The description of all of these concepts except the Hecke growth diagrams can also be found in [13].…”

Linear compactness and combinatorial bialgebras

“…for each increasing tableau T with max(T ) ≤ n. The usefulness of this construction is limited, since it is not known how to easily detect when two increasing tableaux are K-Knuth equivalent. There is an algorithm to compute all K-Knuth classes of words with a given set of letters, however [12].…”