How to construct the lattice of submodules of a multiplicity free module from partial information

Abstract: In general it is a difficult problem to construct the lattice of submodules $L(M )$ given module $M$. In [Sta12] a method is outlined 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 an alternative method based on the composition factors. As input we require a set of submodules $A_1,\ldots, A_n$ whose submodule lattice is known, which contain all composition factors of $M$, and for which all intersections $A_i \cap A_j$ are known. From this we can reconstruct $L(M)$. We illustrate the process for a Verma module M for the Lie superalgebra $\mathfrak{osp}(3, 2)$. In this case, $L(M)$ is isomorphic to the (extended) free distributive lattice of rank 3. This is well-known, but quite complicated lattice. Indeed $M$ has 20 submodules and 8 composition factors, each with multiplicity one.

$\mathbf{Bibliography}$

[Sta12] R. P. Stanley, Enumerative combinatorics. Volume 1, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. MR2868112

rings and algebrasrepresentation theory

Audience: researchers in the topic

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