Kavli Affiliate: David Gross
| First 5 Authors: Felipe Montealegre-Mora, Denis Rosset, Jean-Daniel Bancal, David Gross,
| Summary:
We present a performant and rigorous algorithm for certifying that a matrix
is close to being a projection onto an irreducible subspace of a given group
representation. This addresses a problem arising when one seeks solutions to
semi-definite programs (SDPs) with a group symmetry. Indeed, in this context,
the dimension of the SDP can be significantly reduced if the irreducible
representations of the group action are explicitly known. Rigorous numerical
algorithms for decomposing a given group representation into irreps are known,
but fairly expensive. To avoid this performance problem, existing software
packages — e.g. RepLAB, which motivated the present work — use randomized
heuristics. While these seem to work well in practice, the problem of to which
extent the results can be trusted arises. Here, we provide rigorous guarantees
applicable to finite and compact groups, as well as a software implementation
that can interface with RepLAB. Under natural assumptions, a commonly used
previous method due to Babai and Friedl runs in time O(n^5) for n-dimensional
representations. In our approach, the complexity of running both the heuristic
decomposition and the certification step is O(max{n^3 log n, D d^2 log d}),
where d is the maximum dimension of an irreducible subrepresentation, and D is
the time required to multiply elements of the group. A reference implementation
interfacing with RepLAB is provided.
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