Abstract
We give local axioms that uniquely characterize the crystal-like structure on shifted tableaux developed by the authors and Purbhoo. These axioms closely resemble those developed by Stembridge for type A tableau crystals. This axiomatic characterization gives rise to a new method for proving and understanding Schur $Q$-positive expansions in symmetric function theory, just as the Stembridge axiomatic structure provides for ordinary Schur positivity.
Highlights
Crystal bases were first introduced by Kashiwara [11] in the context of the representation theory of the quantized universal enveloping algebra Uq(g) of a Lie algebra g at q = 0
Their connections to tableau combinatorics, symmetric function theory, and other parts of representation theory have made crystal operators and crystal bases the subject of much recent study. (See [4] for an excellent recent overview of crystal bases.) Crystals have made appearances in geometry, in particular in the structure of the unipotent geometric crystals defined by Berenstein and Kazhdan
A skew shape λ/μ is the difference of the Young diagrams of two partitions λ and μ, where μi λi for all i
Summary
Each connected component C of G has a unique maximal element g∗, with λ = wt(g∗) a partition, and there is a canonical isomorphism C ∼= SSYT(λ, n) of weighted, edge-labeled graphs This result gives a method to show that a symmetric function is Schur-positive: one introduces operators ei, fi on the underlying set, satisfying the local axioms. The raising and lowering operators defined in [8] were introduced to answer geometric questions involving the cohomology of the odd orthogonal Grassmannian H∗(OG(n, V )), where V is a (2n + 1)-dimensional complex vector space with a nondegenerate symmetric form [9] These operators are coplactic, that is, their action is invariant under shifted jeu de taquin slides, and this property essentially determines their action on all shifted skew semistandard tableaux (see Sections 2 and 5 for additional discussion).
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