Crystal structures for double stanley symmetric functions

Hawkes, Graham

doi:10.37236/8872

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Crystal structures for double stanley symmetric functions

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Hawkes, Graham
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.37236/8872
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arXiv:1809.04433.pdf
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Hawkes_Crystal structures for double stanley symmetric functions_2020.pdf
(Publisher version), 366KB

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Citation

Hawkes, G. (2020). Crystal structures for double stanley symmetric functions. The Electronic Journal of Combinatorics, 27(3): P3.15. doi:10.37236/8872.

Cite as: https://hdl.handle.net/21.11116/0000-0006-C3DE-4

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 $C$.