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Mathematics, Combinatorics
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 $C$.
