The character table of the Hecke algebra $H_n(q)$ in terms of traces of products of Murphy operators

Abstract

The traces of the Murphy operators of the Hecke algebra $H_n(q)$, and of products of sets of Murphy operators with non-consecutive indices, can be evaluated by a straightforward recursive procedure. These traces are shown to determine all the reduced traces in this algebra, which, in turn, determine all other traces. To illustrate the procedure we obtain the set of reduced traces for $H_7(q)$ - the lowest order Hecke algebra whose character table has not hitherto been reported. This is preceded by the presentation of an explicit algorithm for the reduction of the trace of an arbitrary element of the Hecke algebra into a linear combination of traces of elements consisting of appropriately defined disjoint cycles; and of a proof, presented in order to make the present article reasonably self-contained, that a reduced trace depends only on the set of lengths of the disjoint cycles that it consists of.