On the existence of a W-graph for an irreducible representation of a Coxeter group

James' conjecture for Hecke algebras of exceptional type, I

“…In fact, Gyoja [24] has shown that every irreducible representation of H K is afforded by a W -graph. We recall: [29].)…”

Computational construction of irreducible W-graphs for types E6 and E7

“…The third argument is a sequence of length equal to the rank of W J describing the embedding of J in S. Thus if this argument is [1,2,3,4,5] (as above), the first simple reflection of W J is identified with the first simple reflection of W , the second with the second, and so on. Given our conventions for numbering the simple roots of D 5 and E 6 , we could use [3,2,1,4,5], [1,2,4,3,6] or [4,2,1,3,6] instead of [1,2,3,4,5] (but there is no point in doing so). For inducing from E 6 to E 7 the third argument of induceWGraph should be [1,2,3,4,5,6] or [1,2,4,3,6,5].…”

“…J:={1,2,3,4,5,6}; graphinfo:=recformat<J,I,edges,basering>; gammaedges:=[{ [2,1], [3,1]}, { [1,1], [5,1], [6,1]}, { [1,1], [4,1], [7,1] Given a W -graph record wg for D 5 , the following command will induce it to E 6 :…”

“…However, Gyoja [A. Gyoja, On the existence of a W -graph for an irreducible representation of a Coxeter group, J. Algebra 86 (1984) 422-438. [3]] proved, by a general argument, that every irreducible representation of a Hecke algebra associated with a finite Coxeter group is given by a W -graph, but this is a pure existence result, and the question remained open of how to construct such W -graphs explicitly. In [R.B.…”

W-graphs for Hecke algebras with unequal parameters

“…A W-graph provides a compact way of providing all the information needed to construct the representation. Moreover, from the work of Gyoja, [11], it is known that any irreducible representation of a Hecke algebra of any finite Weyl group can be afforded by a W -graph.…”