Representation theory of the nonstandard Hecke algebra

TL;DR

This paper explores the representation theory of a specific quotient of the nonstandard Hecke algebra, called the nonstandard Temperley-Lieb algebra, providing a complete classification of its irreducible representations.

Contribution

It offers a complete description of irreducible representations of the nonstandard Temperley-Lieb algebra and shows their restriction properties and basis uniqueness.

Findings

Irreducible representations are fully classified.

Restrictions to smaller algebras are multiplicity-free.

Unique seminormal basis exists up to diagonal transformation.

Abstract

The nonstandard Hecke algebra \check{\mathscr{H}}_r was defined by Mulmuley and Sohoni to study the Kronecker problem. We study a quotient \check{\mathscr{H}}_{r,2} of \check{\mathscr{H}}_r, called the nonstandard Temperley-Lieb algebra, which is a subalgebra of the symmetric square of the Temperley-Lieb algebra TL_r. We give a complete description of its irreducible representations. We find that the restriction of an \check{\mathscr{H}}_{r,2}-irreducible to \check{\mathscr{H}}_{r-1,2} is multiplicity-free, and as a consequence, any \check{\mathscr{H}}_{r,2}-irreducible has a seminormal basis that is unique up to a diagonal transformation.