Character formulae and a realization of tilting modules for $\mathfrak{sl}_2[t]$

TL;DR

This paper characterizes the indecomposable tilting modules for the current algebra of sl_2, showing they are exterior powers of fundamental modules and revealing their structure related to symmetric polynomials.

Contribution

It provides a new realization of tilting modules for sl_2[t] as exterior powers of global Weyl modules and describes their filtration multiplicities.

Findings

Indecomposable tilting modules are exterior powers of fundamental global Weyl modules.

Tilting modules admit a free right action of symmetric polynomials.

Modulo the augmentation ideal, tilting modules relate to dual local Weyl modules.

Abstract

In this paper we study the category of graded modules for the current algebra associated to $\mathfrak{sl}_2$. The category enjoys many nice properties, including a tilting theory which was established in previous work of the authors. We show that the indecomposable tilting modules for $\mathfrak{sl}_2[t]$ are the exterior powers of the fundamental global Weyl module and give the filtration multiplicities in the standard and costandard filtration. An interesting consequence of our result (which is far from obvious from the abstract definition) is that an indecomposable tilting module admits a free right action of the ring of symmetric polynomials in finitely many variables. Moreover, if we go modulo the augmentation ideal in this ring, the resulting $\mathfrak{sl}_2[t]$-module is isomorphic to the dual of a local Weyl module.