No abstract

We construct a simple combinatorially-defined representation of s l 2 \mathfrak {sl}_2 which respects the order structure of the weak order on the symmetric group. This is used to prove that the weak order has the strong Sperner property, and is therefore a Peck poset, solving a problem raised by Björner [Orderings of Coxeter groups, Amer. Math. Soc., Providence, RI, 1984, pp. 175–195]; a positive answer to this question had been conjectured by Stanley [Some Schubert shenanigans, preprint, 2017].

This paper investigates the critical group of a faithful representation of a finite group. It computes the order of the critical group in terms of the character values, and gives some restrictions on its subgroup structure. It also computes the precise structure of the critical group both for the regular representation of any finite group, and for the reflection representation of S n .

No 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].