# Invariant bilinear forms on $W$-graph representations and linear algebra over integral domains

## TL;DR

This paper introduces a new algorithmic method for computing invariant bilinear forms on W-graph representations, enhancing the computational tools available for Lie-theoretic structures like those of type E8.

## Contribution

A novel algorithm for efficiently computing invariant bilinear forms on W-graph representations, facilitating advanced analysis in Lie theory and related algebraic structures.

## Key findings

- Effective computation of invariant bilinear forms achieved
- Algorithm applied successfully to complex Lie-theoretic structures
- Enhanced tools for studying decomposition numbers in Hecke algebras

## Abstract

Lie-theoretic structures of type $E_8$ (e.g., Lie groups and algebras, 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.