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Kostka polynomials and energy functions in solvable lattice models. (English) Zbl 0915.17016
For a partition \(\lambda\) the Kostka number \(K_{\lambda,\mu}\) gives the weight multiplicity of the irreducible \(s\ell_n\)-module \(V(\lambda)\) of weight \(\mu\). It is known from the combinatorics of \(s\ell_n\) representation theory that this Kostka number \(K_{\lambda,\mu}\) is also the transition coefficient between the Schur function \(s_\lambda (x)\) and the symmetric function \(m_\mu (x)\). The Kostka polynomial \(K_{\lambda,\mu}(q)\) is the transition coefficient between the Schur function \(s_\lambda (x)\) and the Hall-Littlewood symmetric function \(P_\mu(x,q)\). In 1978 Lascoux and Schützenberger proved the remarkable fact that \(K_{\lambda,\mu}(q)\) is a polynomial in \(q\) with non-negative integer coefficients. They proved this by showing that \(K_{\lambda,\mu} (q)=\sum q^{c(T)}\), where \(T\) varies over all semi-standard tableaux of shape \(\lambda\) and weight \(\mu\) and \(c(T)\) is an integer valued function, called the “charge” of the tableau \(T\) which is still a mysterious object in combinatorics.
In this paper, the authors use the crystal base theory for the associated affine Lie algebra \(\widehat{s\ell}(n)\) and the corresponding solvable lattice models to give another expression for the Kostka polynomial \(K_{\lambda,\mu}(q)\) in terms of the sum of certain \(q\)-powers where the exponents are given in terms of the energy function. This reproves Lascoux and Schützenberger’s result and establishes the relations between their “charge” and the energy function.
As an application, the authors prove a conjecture of A. N. Kirillov on the expression of the branching coefficient of \(\widehat{s\ell}(n)/s\ell_n\) as a limit of Kostka polynomials.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
05E05 Symmetric functions and generalizations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
82B23 Exactly solvable models; Bethe ansatz
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