# 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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## References

1. 1.

Frame, J.S., Robinson, G., de, B., Thrall, R.M.: The hook graphs of the symmetric groups. Can. J. Math. 6, 316–324 (1954)

2. 2.

Fulton, W.: Young Tableaux: with Applications to Representation Theory and Geometry. Cambridge University Press, New York (1997)

3. 3.

Jayne, R.L., Misra, K.C.: On multiplicities of maximal dominant weights of $$\widehat {sl}(n)$$-modules. Algebr. Represent. Th. 17, 1303–1321 (2014)

4. 4.

Jayne, R.L., Misra, K.C.: Lattice paths, young tableaux, and weight multiplicities. Ann. Combin. 22, 147–156 (2018)

5. 5.

Jimbo, M., Misra, K.C., Miwa, T., Okado, M.: Combinatorics of representations of $$U_{q}\left (\widehat {sl}(n) \right )$$ at q = 0. Commun. Math. Phys. 136, 543–566 (1991)

6. 6.

Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, New York (1990)

7. 7.

Kim, J.S., Lee, K.-H., Oh, S.-J.: Weight multiplicities and Young tableaux through affine crystals Mem. Am. Math. Soc., to appear (2020)

8. 8.

Schensted, C.: Longest increasing and decreasing subsequences. Can. J. Math. 13, 179–191 (1961)

9. 9.

Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, New York (1999)

10. 10.

Tsuchioka, S.: Catalan numbers and level 2 weight structures of $$A^{(1)}_{p-1}$$. RIMS Kǒkyǔroku Bessatsu B11, 145–154 (2009)

11. 11.

Tsuchioka, S., Watanabe, M.: Pattern avoidance seen in multiplicities of maximal weights of affine Lie algebra representations. Proc. Am. Math. Soc. 146, 15–28 (2018)

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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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