Computational construction of irreducible W-graphs for types [formula omitted] and [formula omitted

Abstract

The concept of W-graph was introduced in the influential paper [David Kazhdan, George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184] of Kazhdan and Lusztig. If the underlying Coxeter group is the symmetric group, then Kazhdan and Lusztig showed that every irreducible representation arises from a left cell and, hence, is given by a W-graph. This is the optimal picture that can hope for. For other types of Coxeter groups, the representations arising from left cells are no longer irreducible. 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. Howlett, Yunchuan Yin, Inducing W-graphs, Math. Z. 244 (2003) 415–431] we provided a general method for producing W-graphs, by induction of W-graphs from parabolic subgroups, and then [Yunchuan Yin, Irreducible W-graphs for type D 4 and D 5 , Comm. Algebra 34 (2006) 547–565. [7]] we constructed all the irreducible W-graphs for type D 4 and D 5 by hand calculation. In this paper we will show that the algorithm introduced in [R.B. Howlett, Yunchuan Yin, Inducing W-graphs, Math. Z. 244 (2003) 415–431] is sufficiently powerful to construct explicit W-graphs for all irreducible representations of Hecke algebras of type E 6 and E 7 . The computer algebra system “Magma” was used for the calculations.