## Cubic Hecke algebra on 4 strands

We set R = Z[u,v,w,1/w] and we let H denote the R-algebra defined
as the quotient of the group algebra over R of the (ordinary)
braid group on 4 strands by the relations s^3 - u s^2 + v s - w = 0. In other words,

This object can also be defined as the Hecke algebra
associated to the exceptional complex reflection group G25.
I proved in Cubic Hecke algebras
on at most 5 strands that H is a free R-module
of rank 648, and I provided a basis which is described in the
file baseH4.maple . In this
file, words in the generators are described using the
obvious convention
Inside the files describing the left-regular and right-regular representation we find the matrices mm1,mm2,mm3 of the action of the generators on this basis, as well as the matrices
mm1I,mm2I,mm3I of their inverses.

Matrix models of the irreducible representations of H have been obtained by various people,
including Broué-Malle 1993, Marin 2001 (thesis), Malle-Michel.
In this file, u=a+b+c, v = ab+bc+ac, w = abc, and j is a primitive
3-rd root of 1.

A remarkable quotient of the cubic Hecke algebra on 4 strands gas been studied
in
A
maximal cubic quotient of the braid algebra . The matrices corresponding to the bimodule structure of Q4/Q3u3Q3 over the cubic Hecke algebra on 3 strands can be found in the file
A4tilde.gap.

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