Fermionic formula for double Kostka polynomials

Preprint, Other literature type English OPEN
Liu, Shiyuan (2016)
  • Publisher: Mathematical Society of Japan
  • Journal: (issn: 0025-5645)
  • Related identifiers: doi: 10.2969/jmsj/07017431
  • Subject: 05E10 | crystals | rigged configurations | 82B23 | 05A30 | fermionic formulas | 17B37 | double kostka polynomials | 81R50 | Mathematics - Quantum Algebra
    arxiv: Mathematics::Quantum Algebra | Physics::Space Physics | Physics::Classical Physics

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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