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kalmarek / SymbolicWedderburn.jl

Amazing package to compute decompositions into irreducibles of explicit group representations and the Wedderburn decomposition for endomorphisms thereof!
MIT License
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Further decompose isotypic component #28

Closed blegat closed 3 years ago

blegat commented 3 years ago

If the degree d of the character is larger than 2, we can further decompose the isotypic component into d a direct sum of d sub-components. This can further reduce the SDP as each of these d blocks are equal. Each block will have the dimension given by the corresponding entry in the multiplicities vector.

kalmarek commented 3 years ago

Indeed. Here: https://arxiv.org/pdf/1712.07167.pdf (page 13) we worked out the rank-one projections that produce the decomposition for C₂ ≀ Sym(n) for n≤6. This was done by restricting every character to small subgroup (by composing with the projection onto the characteristic function of the subgroup) and hoping that some of the small subgroups will give you χ·p_H of degree 1. (this brute force search surprisingly works for all characters).

But I don't know how to do it in general (symbolically). I asked (back in 2017) a few experts in the field (representations of finite group) and this seems to be an open problem.

kalmarek commented 3 years ago

partially solved by #38, closing for now