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March 2017 Double Kostka polynomials and Hall bimodule
Shiyuan LIU, Toshiaki SHOJI
Tokyo J. Math. 39(3): 743-776 (March 2017). DOI: 10.3836/tjm/1475723088

Abstract

Double Kostka polynomials $K_{\boldsymbol{\lambda},\boldsymbol{\mu}}(t)$ are polynomials in $t$, indexed by double partitions $\boldsymbol{\lambda}, \boldsymbol{\mu}$. As in the ordinary case, $K_{\boldsymbol{\lambda}, \boldsymbol{\mu}}(t)$ is defined in terms of Schur functions $s_{\boldsymbol{\lambda}}(x)$ and Hall--Littlewood functions $P_{\boldsymbol{\mu}}(x;t)$. In this paper, we study combinatorial properties of $K_{\boldsymbol{\lambda},\boldsymbol{\mu}}(t)$ and $P_{\boldsymbol{\mu}}(x;t)$. In particular, we show that the Lascoux--Sch\"utzenberger type formula holds for $K_{\boldsymbol{\lambda},\boldsymbol{\mu}}(t)$ in the case where $\boldsymbol{\mu} = (-,\mu'')$. Moreover, we show that the Hall bimodule $\mathscr{M}$ introduced by Finkelberg-Ginzburg-Travkin is isomorphic to the ring of symmetric functions (with two types of variables) and the natural basis $\mathfrak{u}_{\boldsymbol{\lambda}}$ of $\mathscr{M}$ is sent to $P_{\boldsymbol{\lambda}}(x;t)$ (up to scalar) under this isomorphism. This gives an alternate approach for their result.

Citation

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Shiyuan LIU. Toshiaki SHOJI. "Double Kostka polynomials and Hall bimodule." Tokyo J. Math. 39 (3) 743 - 776, March 2017. https://doi.org/10.3836/tjm/1475723088

Information

Published: March 2017
First available in Project Euclid: 6 October 2016

zbMATH: 1364.05082
MathSciNet: MR3634291
Digital Object Identifier: 10.3836/tjm/1475723088

Rights: Copyright © 2017 Publication Committee for the Tokyo Journal of Mathematics

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Vol.39 • No. 3 • March 2017
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