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

 

References

[1] A. Bjorner and F. Brenti (2005). Combinatorics of Coxeter groups. Springer-Verlag.

[2] R. Goodman, and N.R. Wallach, Symmetry, Representations and Invariants, Springer, New York,
2009.

[3] P. E. Harris. Computing Weight Multiplicites. 1-25 (2016), preprint.

[4] P. E. Harris, Combinatorial problems related to Kostant‘s weight multiplicity formula. Ph.D. Disser- tation, University of Wisconsin Milwaukee (2012).

[5] B. Kostant,Aformulaforthemultiplicityofaweight,Proc.Natl.Acad.Sci,USA44(1958),588-589.
Published
2018-06-15
How to Cite
HARRIS, Pamela E; LESCINSKY, Haley; MABIE, Grace. Lattice patterns for the support of Kostant’s weight multiplicity formula on $\mathfrak{sl}_3(\mathbb{C})$. 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: 23 oct. 2019.
Section
Articles