The shifted plactic monoid

Abstract

We introduce a shifted analog of the plactic monoid of Lascoux and Schützen-berger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman's mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux-Schützenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.

$\mathfrak{q}$-Crystal Structure on Primed Tableaux and on Signed Unimodal Factorizations of Reduced Words of Type $B$

“…On the other hand, Hawkes et al [13] gave a bijection between the set of semistandard unimodal tableaux and that of primed tableaux of the same shape. The semistandard unimodal tableaux are the same as the semistandard decomposition tableaux originally introduced by Serrano [22] and a bijection between the set of semistandard decomposition tableaux defined in [9,10] and that defined in [22] was established by Choi et al [6]. Combining these results, it follows that primed tableaux form a q-crystal.…”

“…However, the bijection among these two models is established in [6], where the word and the tableau defined in Definition 2.4 are called the reverse hook word and the reverse semistandard decomposition tableau, respectively. The hook word and the semistandard decomposition tableau defined in [4,22] are also called the weakly unimodal word and the semistandard unimodal tableau, respectively [13]. The definition here is more adequate to describe the q(n)-crystal structure because the q(n)-highest and lowest weight vectors take simpler forms.…”

Crystal Analysis of type C Stanley Symmetric Functions

“…In fact, this map is a bijection [16,13]. It follows that the composition SK • HM −1 gives a bijection…”

Crystals and Schur $P$-Positive Expansions

“…Semistandard decomposition tableaux and Schur P -functions. Let us recall the notion of semistandard decomposition tableaux [6,14], which is our main combinatorial object. (1) A word u = u 1 · · · u s in W N is called a hook word if it satisfies u 1 ≥ u 2 ≥ · · · ≥ u k < u k+1 < · · · < u s for some 1 ≤ k ≤ s. In this case, let u↓= u 1 · · · u k be the weakly decreasing subword of maximal length and u↑= u k+1 · · · u s the remaining strictly increasing subword in u.…”

“…be a set of formal commuting variables, and let P λ = P λ (x) be the Schur P -function in x corresponding to λ ∈ P + (see [10]). It is shown in [14] that P λ is given by the weight generating function of SSDT (λ):…”