## “The Adjoint Representation of a Lie Algebra and the Support of Kostant’s Weight Multiplicity Formula.”

Pamela E. Harris | United States Military Academy/Army Research Lab

Friday, February 14 | 2pm | EMS E495

Even though weight multiplicity formulas, such as Kostant’s formula, exist their computational use is extremely cumbersome. In fact, even in cases when the multiplicity is well understood, the number of terms considered in Kostant’s formula is factorial in the rank of the Lie algebra and the value of the partition function is unknown. In this talk, we address the difficult question: What are the contributing terms to the multiplicity of the zero weight in the adjoint representation of a finite dimensional Lie algebra?

We describe and enumerate the cardinalities of these sets (through linear homogeneous recurrence relations with constant coefficients) for the classical Lie algebras of Type B, C, and D, the Type A case was computed by Harris, as part of her dissertation work at UW-Milwaukee. In addition, we compute the cardinality of the set of contributing terms for non-zero weight spaces in the adjoint representation. In the Type B case, the Fibonacci numbers enumerate the cardinality of one such non-zero-weight. We end with some open problems in this area.

This is joint work with Dr. Erik Insko, Florida Gulf Coast University, and Dr. Lauren Kelly Williams, Mercyhurst University.

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