This is not a PLT paper, but I found the programming adventures of mathematicians trying to compute the Kazhdan-Lusztig polynomials for E8 very interesting: The Character Table for E8 by David Vogan. The software part begins on page 9.

I wish he had expanded on the following (emphasis mine in bold):

So late in 2004, Fokko began to add to his software an implementation of the Kazhdan-Lusztig algorithm for real groups. The papers in which this algorithm is formulated are extremely dense, and **written with no consideration for computational practice**. An expert could easily spend many months just to understand the mathematical statements. Fortunately, Jeff Adams had been working on a new formulation of Theorem 6 (parametrizing irreducible representations), growing out of earlier work that he did with Dan Barbasch and me. Jeff's formulation seemed suited to computer implementation; he had been working with Fokko to make it more so.

Over the course of the next year, Fokko understood the Kazhdan-Lusztig algorithm for real groups, **recasting it in the language that he and Jeff had developed**. He wrote clear and efficient code to implement it. In November of 2005 --incredibly soon!-- he finished. Very quickly he and Jeff used the software to compute Kazhdan-Lusztig polynomials (and so character tables) for all of the real forms of F4, E6, and E7, and for the non-split form of E8.

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