Crystal Analysis of type C Stanley Symmetric Functions

Abstract

Combining results of T.K. Lam and J. Stembridge, the type C Stanley symmetric function F C w (x), indexed by an element w in the type C Coxeter group, has a nonnegative integer expansion in terms of Schur functions. We provide a crystal theoretic explanation of this fact and give an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements. . the connections between our crystal operators and those obtained by intertwining crystal operators on words with Haiman's symmetrization of shifted mixed insertion [8, Section 5] and the conversion map [17, Proposition 14] as outlined in Remark 4.11. We thank Toya Hiroshima for pointing out that the definition of

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

“…Note that this definition is different from the one by the convention of Kashiwara. Since we use Kashiwara operators on primed tableaux which are constructed by the anti-Kashiwara convention for the tensor product rule [13], we adopt Definition 2.2.…”

“…Among them, semistandard decomposition tableaux were shown to admit the structure of crystals for the queer Lie superalgebra or simply q-crystal structure by Grantcharov et al [9,10]. 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].…”

“…They further augmented the even Kashiwara operators they gave to the odd Kashiwara operators and constructed a q-crystal structure on semistandard marked shifted tableaux [1,2]. We construct the odd Kashiwara operators on primed tableaux along the lines of [13]. That is, we give the odd Kashiwara operators on primed tableaux by translating the q-crystal structure of words through Serrano's semistandard mixed shifted insertion [22].…”

“…In more detail, the main results of the present paper are summarized as follows. Firstly, as in [13], we employ the semistandard mixed shifted insertion to show that the set of primed tableaux forms a q-crystal and that the odd Kashiwara operators on a primed tableau are independent of a choice of a recording tableau in the above insertion scheme (Theorem 3.2). Secondly, we calculate the highest and lowest weight vectors of primed tableaux by using the rules of ordinary Kashiwara operators on a primed tableau developed in [13] (Theorem 3.3).…”

“…

Crystal basis theory for the queer Lie superalgebra was developed in [9,10], where it was shown that semistandard decomposition tableaux admit the structure of crystals for the queer Lie superalgebra or simply qcrystal structure. In this paper, we explore the q-crystal structure of primed tableaux [13] (semistandard marked shifted tableaux [4]) and that of signed unimodal factorizations of reduced words of type B [13]. We give the explicit odd Kashiwara operators on primed tableaux and the forms of the highest and lowest weight vectors.

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