Great answer! The OP is encouraged to see in particular the Friedman-Hanlon Theorem [Thm 3.2 in Wachs's survey]. It relates the reduced cohomology of $m \times n$ chessboard complexes to irreducible representations of products of symmetric groups $S_m \times S_n$.

Thanks, Mark! I've seen Segal's paper before but didn't think it constituted a roadmap towards actual machine computation. Time to give it another look...

@LennartMeier Sure, eventually I'd like to use a more complicated target category than the 1-truncation, but my impression is that the flat connection/local system case might be easier as a starting point.

@JohnWiltshire-Gordon with $G = \pi_1(X)$, right?

Thanks @davidroberts, now fixed I hope.

I'm not sure what you're asking here. You are free to define "your own Morse function" on a given clique complex, but I have no idea what it means for your Morse function to be "relevant to this process". Which process? Regarding the last sentence, you can use a filtered version of discrete Morse theory to speed up persistent homology computation: link.springer.com/article/10.1007/s00454-013-9529-6. See also Uli Bauer's thesis: webdoc.sub.gwdg.de/diss/2011/bauer_u/bauer_u.pdf

In higher dimensions the answer is easy: alpha and cech are a pain to compute because the smallest-enclosing-ball problem and hyperplane intersections get difficult to compute. On the other hand, Rips is easy to compute but huge (particularly when compared to alpha, which has its dimension bounded by the ambient one).

Thanks, this looks very useful! If I could give another +1 for "Strassen's algorithm reloaded", I would.