The Frobenius transform and the restriction problem

Abstract

Let $\lambda$ and $\mu$ be partitions, and let $n = |\mu|$. A fundamental open problem in algebraic combinatorics is to determine the \emph{restriction coefficients} $r_\lambda^\mu = \dim \operatorname{Hom}{\Sym_n}(V\mu, \mathbb{S}^\lambda \mathbb{C}^n)$, where $V_\mu$ is the Specht module indexed by $\mu$ and $\mathbb{S}^\lambda$ is the Schur functor indexed by $\lambda$. These coefficients are known to be nonnegative integers, but they have no known combinatorial interpretation. The first goal of this dissertation is to explain the significance of restriction coefficients and give an overview of what was previously known about them. The second goal is to prove that these restriction coefficients vanish in certain special cases. The third goal is to provide a combinatorial interpretation of $r_\lambda^\mu$ in the case that $\lambda$ has at most three columns. The fourth goal is to explain the relationship between the restriction coefficients and representations of combinatorial categories. The main tool used to carry out these goals is an abelian group homomorphism~$\Fa$, which we call the \emph{Frobenius transform}, from the ring of symmetric functions to the ring of the symmetric power series. We provide some formulas for $\Fa$ and show how to invert it in special cases.