  A fundamental open problem in algebraic combinatorics is to find a positive
combinatorial formula for Kronecker coefficients, which are multiplicities of
the decomposition of the tensor product of two \S_r-irreducibles into
irreducibles. Mulmuley and Sohoni attempt to solve this problem using canonical
basis theory, by first constructing a nonstandard Hecke algebra B_r, which,
though not a Hopf algebra, is a u-analogue of the Hopf algebra \CC \S_r in some
sense (where u is the Hecke algebra parameter). For r=3, we study this
Hopf-like structure in detail. We define a nonstandard Hecke algebra
\bar{\H}^{(k)}_3 \subseteq \H_3^{\tsr k}, determine its irreducible
representations over \QQ(u), and show that it has a presentation with a
nonstandard braid relation that involves Chebyshev polynomials evaluated at
\frac{1}{u + u^{-1}}. We generalize this to Hecke algebras of dihedral groups.
We go on to show that these nonstandard Hecke algebras have bases similar to
the Kazhdan-Lusztig basis of \H_3 and are cellular algebras in the sense of
Graham and Lehrer.

Representation Theory