Abstract
We relate the combinatorial definitions of the type $A_n$ and type $C_n$ Stanley symmetric functions, via a combinatorially defined "double Stanley symmetric function," which gives the type $A$ case at $(\mathbf{x},\mathbf{0})$ and gives the type $C$ case at $(\mathbf{x},\mathbf{x})$. We induce a type $A$ bicrystal structure on the underlying combinatorial objects of this function which has previously been done in the type $A$ and type $C$ cases. Next we prove a few statements about the algebraic relationship of these three Stanley symmetric functions. We conclude with some conjectures about what happens when we generalize our constructions to type $C$.
Highlights
Introduction and NotationIn this paper we will relate the combinatorial definitions of the type An ([Sta84]) and type Cn+1 [BH95], [FK96] Stanley symmetric functions
Crystal analysis for the stable limit of Schubert polynomials is carried out [Len04] by considering a crystal structure on the underlying combinatorial objects of rc graphs
In the paper we will carry out this procedure for the double Stanley symmetric function by considering a crystal sructure for the underlying objects of reduced signed increasing factorizations
Summary
In this paper we will relate the combinatorial definitions of the type An ([Sta84]) and type Cn+1 [BH95], [FK96] Stanley symmetric functions. In the paper we will carry out this procedure for the double Stanley symmetric function by considering a crystal sructure for the underlying objects of reduced signed increasing factorizations. Given the relations above one can define two types of symmetric functions, the electronic journal of combinatorics 27(3) (2020), #P3.15 indexed, respectively, by elements of An and Cn+1. The resulting system is not Coxeter, for instance, the relations imply that s−i = si holds, 1 so the generating set is obviously not minimal In this setting, a reduced word for ω is an expression, u, for ω using the generators s−n, . We define the double Stanley symmetric polynomial in k variables for ω ∈ Cn+1 to be: Fωd(x, y) =.
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