How to Construct the Lattice of Submodules of a Multiplicity free Module from Partial Information

TL;DR

This paper presents a method to construct the lattice of submodules of a multiplicity free module using partial information about known submodules and their composition factors, with applications to specific Lie superalgebra modules.

Contribution

It introduces a modified Stanley's method tailored for multiplicity free modules, enabling lattice reconstruction from partial submodule data.

Findings

Lattice of submodules of certain Verma modules is isomorphic to a free distributive lattice.

Method simplifies the construction of submodule lattices for multiplicity free modules.

Application demonstrated for modules over the Lie superalgebra osp(3,2).

Abstract

In general it is a difficult problem to construct the lattice of submodules $L(M)$ of a given module $M$. In \cite{St} R. P. Stanley outlined a method for constucting a distributive lattice from a knowledge of its join irreducibles. However it is not an easy task to identify all join irreducible submodules of a given module. In the case of a multiplicity free module $M$ we present a modifiiction of Stanley's method based on the composition factors of $M$. As input we require a set of submodules $A_1,\ldots , A_n$ whose submodule lattices are known and which contain all composition factors of $M$. From this we can reconstruct $L(M)$. We illustrate the process for a family of Verma modules $M(\gl_n)$, with $n $ a positive integer, for the Lie superalgebra $\osp(3,2)$. We show that for $n\ge 2$, $L(M(\gl_n))$ is isomorphic to the (extended) free distributive lattice of rank 3.