650
D. Rosset et al.
algebraic techniques. However, equivalent irreps are not always grouped nor put in
the same basis, and AREP works only over the complex numbers.
Character Table Approaches It is possible to use the character table [42–44] of a
finite group to decompose a generic representation ρ: by expanding the character
χ : g → tr[ρ g ] as a linear combination of irreducible characters, one can construct
a block-diagonal representation ρ similar to ρ. However, this approach does not
provide directly the change of basis matrix. GAP itself can compute the character
table and irreducible representations, but does not provide the change of basis
matrix that reveals the irreducible decomposition. 7 These first two approaches
have the advantage of providing exact algebraic solutions. However, they do not
apply to compact groups, and depend on a large library of algorithms and data
tables. In comparison, RepLAB is lightweight and readily integrates with the
MATLAB/Octave ecosystem related to semidefinite optimization.
Approaches Based on ∗-Algebra Decomposition By observing that the commutant
algebra C ρ is a ∗-algebra [36], it can be decomposed using the Artin–Wedderburn
theorem; this has been discussed several times in the context of invariant SDP
programs [35, 36, 45]. We found an incomplete implementation of the algorithm
of [35] in the software package NCSOSTools [46]. However, the paper [35]
prescribes the use of exotic matrix decomposition techniques, some of which do
not have implementations available.
The approach used in RepLAB directly inspired by this last class of approaches;
we will describe our technical improvements in a future publication. Note that
RepLAB does not aim to replace a Computer Algebra System such as GAP System;
rather, it focuses on the numerical symmetrization of optimization problems.
4 Conclusion
We presented a toolbox to numerically decompose arbitrary finite dimensional
representation of compact groups. Surprisingly, the user needs to provide only
little information about the group and its representation, and there is no need to
compute much structure to accomplish our task. The current unoptimized code is
able to deal with representations of size up to ∼10,000. We foresee RepLAB having
impact in two ways. The first one is reducing the computational cost of solving
SDPs, thus expanding the applicability of a wide variety of quantum information
methods. The second one is pedagogical: by delegating all computations to the
software, a hands-on approach to representation theory can be taught, focusing
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.
D. Rosset et al.
algebraic techniques. However, equivalent irreps are not always grouped nor put in
the same basis, and AREP works only over the complex numbers.
Character Table Approaches It is possible to use the character table [42–44] of a
finite group to decompose a generic representation ρ: by expanding the character
χ : g → tr[ρ g ] as a linear combination of irreducible characters, one can construct
a block-diagonal representation ρ similar to ρ. However, this approach does not
provide directly the change of basis matrix. GAP itself can compute the character
table and irreducible representations, but does not provide the change of basis
matrix that reveals the irreducible decomposition. 7 These first two approaches
have the advantage of providing exact algebraic solutions. However, they do not
apply to compact groups, and depend on a large library of algorithms and data
tables. In comparison, RepLAB is lightweight and readily integrates with the
MATLAB/Octave ecosystem related to semidefinite optimization.
Approaches Based on ∗-Algebra Decomposition By observing that the commutant
algebra C ρ is a ∗-algebra [36], it can be decomposed using the Artin–Wedderburn
theorem; this has been discussed several times in the context of invariant SDP
programs [35, 36, 45]. We found an incomplete implementation of the algorithm
of [35] in the software package NCSOSTools [46]. However, the paper [35]
prescribes the use of exotic matrix decomposition techniques, some of which do
not have implementations available.
The approach used in RepLAB directly inspired by this last class of approaches;
we will describe our technical improvements in a future publication. Note that
RepLAB does not aim to replace a Computer Algebra System such as GAP System;
rather, it focuses on the numerical symmetrization of optimization problems.
4 Conclusion
We presented a toolbox to numerically decompose arbitrary finite dimensional
representation of compact groups. Surprisingly, the user needs to provide only
little information about the group and its representation, and there is no need to
compute much structure to accomplish our task. The current unoptimized code is
able to deal with representations of size up to ∼10,000. We foresee RepLAB having
impact in two ways. The first one is reducing the computational cost of solving
SDPs, thus expanding the applicability of a wide variety of quantum information
methods. The second one is pedagogical: by delegating all computations to the
software, a hands-on approach to representation theory can be taught, focusing
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.