TY - JOUR
AU - Harris, Pamela E
AU - Lescinsky, Haley
AU - Mabie, Grace
PY - 2018
TI - Lattice patterns for the support of Kostantâ€™s weight multiplicity formula on $\mathfrak{sl}_3(\mathbb{C})$
JF - Minnesota Journal of Undergraduate Mathematics; Vol 4 No 1
KW -
N2 - 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.
UR - https://mjum.math.umn.edu/index.php/mjum/article/view/45