Abstract
We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of `doubled crystal' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are the (skew) Schur Q functions.
Highlights
A crystal base is a set B along with certain raising and lowering operators Ei, Fi : B → B ∪ {∅}, functions φi, εi : B → Z ∪ {−∞}, and a weight map wt : B → Λ where Λ is a weight lattice of some Lie type
Crystal bases were first introduced by Kashiwara [13] in the context of the representation theory of the quantized universal enveloping algebra Uq(g) of a Lie algebra g at q = 0
Two standard skew shifted tableaux are said to be dual equivalent if their shapes transform the same way under any sequence of jeu de taquin slides. (See [1, 11] for more in-depth discussions of dual equivalence.) We extend this notion to semistandard shifted tableaux by defining two tableaux to be dual equivalent if and only if their standardizations are, as in [18]
Summary
A crystal base is a set B along with certain raising and lowering operators Ei, Fi : B → B ∪ {∅}, functions φi, εi : B → Z ∪ {−∞}, and a weight map wt : B → Λ where Λ is a weight lattice of some Lie type. Crystal bases were first introduced by Kashiwara [13] in the context of the representation theory of the quantized universal enveloping algebra Uq(g) of a Lie algebra g at q = 0. Since their connections to tableau combinatorics, symmetric function theory, and other aspects of representation theory have made crystal operators and crystal bases the subject of much recent study. The type A crystal base theory can be described entirely in terms of semistandard Young tableaux. Combinatorial crystals, shifted Young tableaux, symmetric function theory, orthogonal Grassmannian
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