The Pieri Rule at Infinity

TL;DR

This paper investigates tensor products of infinite-dimensional modules over rak() and reveals their indecomposability, simple constituents, and filtration structures, extending classical Pieri rule insights to an infinite-dimensional setting.

Contribution

It characterizes the structure of tensor products of rak()-modules, introducing a linkage filtration and analyzing semisimplicity and indecomposability in the infinite-dimensional case.

Findings

Tensor products are generally indecomposable, except in trivial cases.

A linkage filtration helps understand the relationships between simple constituents.

The socle and radical filtrations are explicitly computed, and conditions for rigidity are established.

Abstract

We study the structure of tensor products of $\mathfrak{gl}(\infty) = \varinjlim \mathfrak{gl}(n)$-modules $\mathbf L(\mathbf \lambda) \otimes \mathbf F$ where $\mathbf L(\mathbf \lambda)$ is a simple integrable highest weight module and $\mathbf F$ is a simple integrable weight multiplicity-free module. Both $\mathbf L(\mathbf \lambda)$ 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 $\mathbf M:= \mathbf L(\mathbf \lambda) \otimes \mathbf F$ 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.