Fermionic formula for double Kostka polynomials. (English) Zbl 1429.17016

Summary: 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 \(K_{\lambda,\mu}(t)\), indexed by two double partitions \(\lambda,\mu,\) are polynomials in \(t\) introduced as a generalization of Kostka polynomials.
In the present paper, we consider \(K_{\lambda,\mu}(t)\) in the special case where \(\mu=(-,\mu'')\). We formulate a 1D sum and a fermionic formula for \(K_{\lambda,\mu}(t)\), as a generalization of the case of ordinary Kostka polynomials. Then we prove an analogue of the \(X=M\) conjecture.


17B37 Quantum groups (quantized enveloping algebras) and related deformations
05E10 Combinatorial aspects of representation theory
82B23 Exactly solvable models; Bethe ansatz
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
05A30 \(q\)-calculus and related topics
Full Text: DOI arXiv


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