Kostant's Formula and Parking Functions: Combinatorial Explorations

Abstract

In the first part of this work, we focus on Kostant’s weight multiplicity formula. 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). This is 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 the Weyl alternation sets are order ideals in the weak Bruhat order of the Weyl group, applying structure to where there was none. Turning our attention to specifically the Lie algebra of type A, we prove that the Weyl alternation set 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. Turning our focus to when µ is a negative root of sl_{r+1}(C), we give a complete characterization of the Weyl alternation sets and show that the cardinality of these sets satisfies a two-term recurrence relation involving Fibonacci numbers. In the second part of this work, we focus on a combinatorial family of tuples called parking functions. We present results related to a class of parking functions in which cars have varying lengths and provide enumerative formulas. Lastly, we consider different statistics on specific parking functions and find that they are related to Catalan and Fine numbers.