# Lattice patterns for the support of Kostant’s weight multiplicity formula on $\mathfrak{sl}_3(\mathbb{C})$

### Abstract

The multiplicity of a weight in a finite-dimensional irreducible representation of the Lie algebra $\mathfrak{sl}_{3} (\mathbb{C}) $

can be computed via Kostant's weight multiplicity formula. This formula consists of an alternating sum over a finite group and involves a partition function. Our main result describes the terms that contribute nonzero values to this formula, as, in practice, most terms in the sum contribute a value of zero. By taking a geometric approach, we provide concrete visualizations of these sets for all pairs of integral weights $\lambda$ and $\mu$ of $\mathfrak{sl}_3(\mathbb{C})$ and show that the diagrams associated to our main result present new and surprising symmetry.

### References

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[5] B. Kostant,Aformulaforthemultiplicityofaweight,Proc.Natl.Acad.Sci,USA44(1958),588-589.

**Minnesota Journal of Undergraduate Mathematics**, [S.l.], v. 4, n. 1, june 2018. ISSN 2378-5810. Available at: <https://mjum.math.umn.edu/index.php/mjum/article/view/45>. Date accessed: 14 july 2020.