Abstract
We present a MATLAB/Octave toolbox to decompose finite dimensional representations of compact groups. Surprisingly, little information about the group and the representation is needed to perform that task. We discuss applications to semidefinite programming.
Keywords
- Representation theory
- Compact groups
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Notes
- 1.
- 2.
RepLAB works with nonunitary representations of compact groups too as they can have numerical advantages. We keep this presentation simple by assuming unitarity.
- 3.
This saves on memory and CPU time requirements. Consider a SDP problem \(\min _X \operatorname {tr} [ C X ]\) such that X is SDP and \(\operatorname {tr}[ A_i X ] = b_i\) for i = 1, …, m. Assume X has dimension n × n with blocks of size n i so that n = n 1 + … + n I. The complexity of standard interior point primal-dual methods is as follows. For CPU time [27]: when m ≫ n, the factoring of the Schur complement matrix dominates in \(\mathcal {O} (m^3)\). When mλn (which we observe is the common case), Cholesky factorizations and eigenvalue computations usually dominate, in \(\mathcal {O} ((n_1)^3 + \ldots (n_I)^3)\). For memory: the problem data scales in \(\mathcal {O} (mn^2)\) in the worst-case, but often less due to sparsity. The Schur complement matrix requires \(\mathcal {O} (m^2)\) storage, and the matrices X and χ require storage in \(\mathcal {O} ((n_1)^2 + \ldots + (n_I)^2)\). When using our technique, the block-diagonalization of a SDP of size n × n produces a SDP with blocks of size \(n^{\prime }_i = M_i\).
- 4.
For example, every element of the symmetric group S D can be written uniquely as a product of powers of the cycles (1, 2), (1, 2, 3), …, (1, …, D), reducing the computational effort from \(\mathcal {O}(|G|) = \mathcal {O}(D!)\) to \(\mathcal {O}(D^2)\) image computations.
- 5.
Note that the same proposition can be adapted to decide whether two irreducible representations of G, \(\sigma ^1: G \rightarrow \mathcal {U}(n_1)\) and \(\sigma ^2: G \rightarrow \mathcal {U}(n_2)\) are equivalent, and computing the change of basis matrix between them. This problem, considered in [39] for finite groups only, can be solved for compact groups by applying the proposition above to \(\rho : g \mapsto \rho _g = \sigma ^1_g \oplus \sigma ^2_g\).
- 6.
Usually, σ is a subrepresentation of some ρ, so that we sample from C ρ and restrict.
- 7.
Note that for groups of small order, the approach discussed in [26] could give the change of basis matrix. But note that the authors of [23] remarked that the exact algorithms of GAP were sometimes slow and restricted their decompositions to groups of order <100 in their symmetrization of conformal bootstrap.
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Acknowledgements
We acknowledge useful discussions with David Gross, Elie Wolfe, and Markus Heinrich. This research was supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade. This publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. FMM was funded by the DFG project number 4334.
Note Added
During the review of this manuscript, we became aware of the publication of the package RepnDecomp [47], integrated in the release 4.11.0 of GAP (March 2020).
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Rosset, D., Montealegre-Mora, F., Bancal, JD. (2021). RepLAB: A Computational/Numerical Approach to Representation Theory. In: Paranjape, M.B., MacKenzie, R., Thomova, Z., Winternitz, P., Witczak-Krempa, W. (eds) Quantum Theory and Symmetries. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-55777-5_60
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