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Weight multiplicity polynomials for affine Kac-Moody algebras of type \(A_ r^{(1)}\). (English) Zbl 0862.17016

For the affine Kac-Moody algebras \(X_r^{(1)}\) it has been conjectured by the first author and S. N. Kass [Queen’s Papers Pure Appl. Math. 94, 1-12 (1994; Zbl 0818.17027)] that for fixed dominant weight \(\lambda\) and \(\mu\) the multiplicity \(\dim L (\lambda)_\mu\) of the weights \(\mu\) in the irreducible \(X_r^{(1)}\)-module \(L(\lambda)\) of highest weight \(\lambda\) is a polynomial in \(r\) which depends on the type \(X\) of the algebra.
In the paper under review the authors provide a precise conjecture for the degree of that polynomial for the algebras \(A_r^{(1)}\). To offer evidence for this conjecture, the authors prove it for all dominant weights \(\lambda\) and all weights \(\mu\) of depth \(\leq 2\), that is, for all weights of the form \(\mu = \nu\), \(\nu-\delta\) and \(\nu - 2 \delta\). For those weights the multiplicities \(\dim L(\lambda)_\mu\) are explicitly expressed as polynomials in \(r\) with coefficients involving Kostka numbers. These expressions are obtained by using the results of the second author [Duke Math. J. 74, 635-666 (1994; Zbl 0823.17031)] and combinatorial identities involving various Kostka numbers.
Reviewer: M.Primc (Zagreb)

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
05A19 Combinatorial identities, bijective combinatorics
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References:

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