Abstract
Combining results of T.K. Lam and J. Stembridge, the type $C$ Stanley symmetric function $F_w^C(\mathbf{x})$, 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.
Highlights
Schubert polynomials of type B and type C were independently introduced by Billey and Haiman [1] and Fomin and Kirillov [6]
It follows that Stanley symmetric functions of type B have a positive integer expansion in terms of Schur functions
In [15], this was exploited to provide a combinatorial interpretation in terms of highest weight crystal elements of the coefficients in the Schur expansion of Stanley symmetric functions in type
Summary
Schubert polynomials of type B and type C were independently introduced by Billey and Haiman [1] and Fomin and Kirillov [6]. Stanley symmetric functions [18] are stable limits of Schubert polynomials, designed to study properties of reduced words of Coxeter group elements. Stembridge [19] proved that the P-Schur functions expand positively in terms of Schur functions Combining these two results, it follows that Stanley symmetric functions of type B (and type C) have a positive integer expansion in terms of Schur functions. Schur functions sλ(x), indexed by partitions λ, are ubiquitous in combinatorics and representation theory They are the characters of the symmetric group and can be interpreted as characters of type A crystals. In [15], this was exploited to provide a combinatorial interpretation in terms of highest weight crystal elements of the coefficients in the Schur expansion of Stanley symmetric functions in type. Appendices A and B are reserved for the proofs of Theorems 4.3 and 4.5
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