Multiplicities of Some Maximal Dominant Weights of the \(\widehat {s\ell }(n)\)-Modules V (kΛ0)

Abstract

For n ≥ 2 consider the affine Lie algebra \(\widehat {s\ell }(n)\) with simple roots {αi∣0 ≤ in − 1}. Let \(V(k{\Lambda }_{0}), k \in \mathbb {Z}_{\geq 1}\) denote the integrable highest weight \(\widehat {s\ell }(n)\)-module with highest weight kΛ0. It is known that there are finitely many maximal dominant weights of V (kΛ0). Using the crystal base realization of V (kΛ0) and lattice path combinatorics we examine the multiplicities of a large set of maximal dominant weights of the form \(k{\Lambda }_{0} - \lambda ^{\ell }_{a,b}\) where \( \lambda ^{\ell }_{a,b} = \ell \alpha _{0} + (\ell -b)\alpha _{1} + (\ell -(b+1))\alpha _{2} + {\cdots } + \alpha _{\ell -b} + \alpha _{n-\ell +a} + 2\alpha _{n - \ell +a+1} + {\ldots } + (\ell -a)\alpha _{n-1}\), and ka + b, \(a,b \in \mathbb {Z}_{\geq 1}\), \(\max \limits \{a,b\} \leq \ell \leq \left \lfloor \frac {n+a+b}{2} \right \rfloor -1 \). We obtain two formulae to obtain these weight multiplicities - one in terms of certain standard Young tableaux and the other in terms of certain pattern-avoiding permutations.

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Correspondence to Rebecca L. Jayne.

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KCM: partially supported by Simons Foundation grant #636482

Presented by: Pramod Achar

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Jayne, R.L., Misra, K.C. Multiplicities of Some Maximal Dominant Weights of the \(\widehat {s\ell }(n)\)-Modules V (kΛ0). Algebr Represent Theor (2021). https://doi.org/10.1007/s10468-021-10031-3

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Keywords

  • Affine Lie algebra
  • Crystal base
  • Lattice path
  • Young tableau
  • Pattern-avoiding permutation

Mathematics Subject Classification (2010)

  • Primary 17B67
  • 17B37
  • 17B10
  • Secondary 05A05
  • 05E10
  • 05A17