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Lie-theoretic structures of type E8 (e.g., Lie groups and algebras, Iwahori–Hecke algebras and Kazhdan–Lusztig cells, …) are considered to serve as a “gold standard” when it comes to judging the effectiveness of a general algorithm for solving a computational problem in this area. Here, we address a problem that occurred in our previous work on decomposition numbers of Iwahori–Hecke algebras, namely, the computation of invariant bilinear forms on so-called W-graph representations. We present a new algorithmic solution which makes it possible to produce and effectively use the main results in further applications.

Invariant Bilinear Forms on W-Graph Representations and Linear Algebra Over Integral Domains