A semisimple series for $q$-Weyl and $q$-Specht modules

TL;DR

This paper extends the understanding of semisimple series for quantum Weyl and Specht modules, providing explicit filtrations and multiplicities for certain types and parameters, with implications for classical algebraic structures.

Contribution

It generalizes semisimple series results to all positive integers e for type A and certain D types, and applies these to q-Specht modules and classical algebraic modules.

Findings

Explicit semisimple series for quantum Weyl modules in type A and D.

Computable multiplicities of irreducible constituents in filtrations.

Application to semisimple series on q-Specht modules and classical modules.

Abstract

In a previous paper, the authors studied the radical filtration of a Weyl module $\Delta_\zeta(\lambda)$ for quantum enveloping algebras $U_\zeta(\overset\circ{\mathfrak g})$ associated to a finite dimensional complex semisimple Lie algebra $\overset\circ{\mathfrak g}$. There $\zeta^2=\sqrt[e]{1}$ and $\lambda$ was, initially, required to be $e$-regular. Some additional restrictions on $e$ were required---e.g., $e>h$, the Coxeter number, and $e$ odd. Translation to a facet gave an explicit semisimple series for all quantum Weyl modules with singular, as well as regular, weights. That is, the sections of the filtration are explicit semisimple modules with computable multiplicities of irreducible constituents. However, in the singular case, the filtration conceivably might not be the radical filtration. This paper shows how a similar semisimple series result can be obtained for all positive integers $e$ in case $\overset\circ{\mathfrak g}$ has type $A$, and for all positive integes $e\geq 3$ in type $D$. One application describes semisimple series (with computable multiplicities) on $q$-Specht modules. We also discuss an analogue for Weyl modules for classical Schur algebras and Specht modules for symmetric group algebras in positive characteristic $p$. Here we assume the James Conjecture and a version of the Bipartite Conjecture.