Lattice patterns for the support of Kostant’s weight multiplicity formula on $\mathfrak{sl}_3(\mathbb{C})$
Pamela E Harris
Williams College
Haley Lescinsky
Williams College
Grace Mabie
Williams College
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.
Author Biography
Haley Lescinsky, Williams College
