Indecomposable tilting modules for the blob algebra

TL;DR

This paper constructs indecomposable tilting modules for the blob algebra in characteristic zero, revealing their structure and connection to Kazhdan-Lusztig polynomials, especially in the doubly critical case.

Contribution

It provides a detailed construction and classification of indecomposable tilting modules for the blob algebra in the doubly critical case over a field of characteristic zero.

Findings

Indecomposable tilting modules are either projective or extensions of simple modules by projective modules.

Every indecomposable tilting module is a submodule of a maximal highest weight tilting module.

Graded Weyl multiplicities are given by inverse Kazhdan-Lusztig polynomials of type A˜1.

Abstract

The blob algebra is a finite-dimensional quotient of the Hecke algebra of type $B$ which is almost always quasi-hereditary. We construct the indecomposable tilting modules for the blob algebra over a field of characteristic $0$ in the doubly critical case. Every indecomposable tilting module of maximal highest weight is either a projective module or an extension of a simple module by a projective module. Moreover, every indecomposable tilting module is a submodule of an indecomposable tilting module of maximal highest weight. We conclude that the graded Weyl multiplicities of the indecomposable tilting modules in this case are given by inverse Kazhdan-Lusztig polynomials of type $\tilde{A}_1$.