Kostant’s Formula and Parking Functions: Combinatorial Explorations

Kimberly Harry
University of Wisconsin-Milwaukee

We let L(λ) denote the irreducible highest weight representation of the classical simple Lie algebra g with highest weight λ. Kostant’s weight multiplicity formula gives a way to compute the multiplicity of a weight µ in L(λ), denoted m(λ, µ), via an alternating sum over the Weyl group whose terms involve the Kostant partition function. The Weyl alternation set A(λ, µ) is the set of Weyl group elements that contribute nontrivially to the multiplicity m(λ, µ). We prove that Weyl alternation sets are order ideals in the weak Bruhat order of the Weyl group. Specializing to the Lie algebra of type A, we prove that the Weyl alternation sets A(˜α, µ), where ˜α is the highest root of sl_{r+1}(C) and µ is a positive root is a product of Fibonacci numbers. Using this result, we show that the q-multiplicity of the positive root in the representation L(˜α) is precisely a power of q. We give a complete characterization of the Weyl alternation sets A(˜α, µ), where µ is now a negative root of sl_{r+1}(C). We also show that the cardinality of these Weyl alternation sets satisfies a two-term recurrence relation involving Fibonacci numbers. Time permitting I will present further results related to collaborative projects I have contributed to during my years at UWM.

Advisor: Pamela E. Harris

Committee Members:
Profs. Jeb Willenbring, Kevin McLeod, Gabriella Pinter, and Jonah Gaster