The Pieri Rule at Infinity

Abstract

We study the structure of tensor products of $\mathfrak{gl}(\infty )=\underrightarrow{\lim }\mathfrak{gl}(n)$-modules $\mathbf{L}(\lambda )\otimes \mathbf{F}$ where $\mathbf{L}(\lambda )$ is a simple integrable highest weight module and $\mathbf{F}$ is a simple integrable weight multiplicity-free module. Both $\mathbf{L}(\lambda )$ and $\mathbf{F}$ are infinite dimensional, in particular $\mathbf{F}$ can be a Fock module. Similar tensor products of $\mathfrak{gl}(n)$-modules are semisimple and their simple constituents are described by the classical Pieri rule. We prove that a $\mathfrak{gl}(\infty )$-module $\mathbf{M}:=\mathbf{L}(\lambda )\otimes \mathbf{F}$ is semisimple only in relatively trivial cases, and is indecomposable otherwise. Our main results are a description of the simple constituents of $\mathbf{M}$, and the construction of a linkage filtration on $\mathbf{M}$ that provides information on when two simple constituents of $\mathbf{M}$ are linked. Using the linkage filtration, we compute the socle and radical filtrations of $\mathbf{M}$, and determine when $\mathbf{M}$ is rigid.