Representation Theory of the Nonstandard Hecke Algebra

Abstract

The nonstandard Hecke algebra $\check {\mathcal {H}}_{r}$ was defined by Mulmuley and Sohoni to study the Kronecker problem. We study a quotient $\check {\mathcal {H}}_{r,2}$ of $\check {\mathcal {H}}_{r}$, 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 irreducible $\check {\mathcal {H}}_{r,2}$-module to $\check {\mathcal {H}}_{r-1,2}$ is multiplicity-free, and as a consequence, any irreducible $\check {\mathcal {H}}_{r,2}$-module has a seminormal basis that is unique up to a diagonal transformation.