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January, 2018 Fermionic formula for double Kostka polynomials
Shiyuan LIU
J. Math. Soc. Japan 70(1): 283-324 (January, 2018). DOI: 10.2969/jmsj/07017431


The $X=M$ conjecture asserts that the $1D$ sum and the fermionic formula coincide up to some constant power. In the case of type $A,$ both the $1D$ sum and the fermionic formula are closely related to Kostka polynomials. Double Kostka polynomials $K_{{\boldsymbol\lambda},{\boldsymbol\mu}}(t),$ indexed by two double partitions ${\boldsymbol\lambda},{\boldsymbol\mu},$ are polynomials in $t$ introduced as a generalization of Kostka polynomials. In the present paper, we consider $K_{{\boldsymbol\lambda},{\boldsymbol\mu}}(t)$ in the special case where ${\boldsymbol\mu}=(-,\mu'')$. We formulate a $1D$ sum and a fermionic formula for $K_{{\boldsymbol\lambda},{\boldsymbol\mu}}(t),$ as a generalization of the case of ordinary Kostka polynomials. Then we prove an analogue of the $X=M$ conjecture.


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Shiyuan LIU. "Fermionic formula for double Kostka polynomials." J. Math. Soc. Japan 70 (1) 283 - 324, January, 2018.


Published: January, 2018
First available in Project Euclid: 26 January 2018

zbMATH: 06859853
MathSciNet: MR3750277
Digital Object Identifier: 10.2969/jmsj/07017431

Primary: 17B37
Secondary: 05A30 , 05E10 , 81R50 , 82B23

Keywords: crystals , double kostka polynomials , fermionic formulas , rigged configurations

Rights: Copyright © 2018 Mathematical Society of Japan


Vol.70 • No. 1 • January, 2018
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