## Abstract

For *n* ≥ 2 consider the affine Lie algebra \(\widehat {s\ell }(n)\) with simple roots {*α*_{i}∣0 ≤ *i* ≤ *n* − 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 *k*Λ_{0}. It is known that there are finitely many maximal dominant weights of *V* (*k*Λ_{0}). Using the crystal base realization of *V* (*k*Λ_{0}) 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 *k* ≥ *a* + *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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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* (*k*Λ_{0}).
*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