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@DaveBenson excuse me if I'm being dense but I'm not sure how to use this. I have a list of matrices $\partial_n : \mathbb{F}_2^{a_n} \to \mathbb{F}_2^{a_{n+1}}$ with $\partial_n \partial_{n+1}=0$; how would I transform these into integer matrices?
@RobPratt I'm interested to see how integer linear programming can be applied here. I think the problem can re-stated as finding linear combinations of the rows of $H$ with the least maximum weight...you'd have to add that the rows are independent which will complicate things...I can ask a post a separate question is this is different from OP's
it should be straight forward to automate finding such expressions : GAP for example can construct all these representations explicitly and can give decompositions of symmetric and exterior products..
@Friedrich: my knowledge of Kac-Moody algebras is that they correspond to root systems with signature (++...++0); these have been classified and I'm aware of the classification. Do you have a more specific reference for Kac's book? especially if it handles other signatures as well
but I would think at some point the signature enters the picture...for example in 4 dimensional real space all reflection groups have been classified for signature (++++)...I don't think the same has been done for (+++-) or (++--) case
I'm thinking along the lines of root systems arising from finite real reflection groups where the reflections $\phi_w : v \to v - 2 w(v .w)/(w.w)$ are with with respect to a non euclidean inner product $(x.y)$
in the special case where the field is GF(2) and there are no numerical issues, it still seems that we still need permutation matrices in case the pivot is not in the right place...this would add $n-1$ matrices to the $n(n-1)$ matrices above; I think the permutation can be absorbed into one of the $E_{ij}$'s or as a slight modification to the Frobenius matrices but I haven't worked that out yet
@Rodrigo...this is very nice. A minor point : wouldn't there be some permutation matrices in the product too? or are you combining these with the $E_{ij}$ factors?
as a special case, matrices that are known to be the inverse of a sparse matrix are of interest...also the field could be GF(2). I don't know if these additional restrictions change the problem much
