UC Berkeley

Catabolism is an operation on tableaux originally defined by Lascoux. Though the operation is not difficult to compute, many of its properties remain elusive. The combinatorics of catabolism has deep ties to various generalizations and subspaces of the Type A coinvariant ring and their structures as graded symmetric group representations. This in turn gives connections to the corresponding q-symmetric functions that give the graded Frobenius characters of these spaces. In this dissertation, we explore these connections from both the combinatorial and algebraic perspective. We use catabolizability to construct bases for different families of generalized coinvariant rings. Our constructions give bridges between the combinatorics of catabolism and the representation theoretic structures: in particular, we recover combinatorial formulas for the Schur expansions of their graded Frobenius characters. We also study the combinatorics of catabolism directly in an attempt to identify submodules of the coinvariant ring with Frobenius character given by k-Schur functions.