RepLAB: A Computational/Numerical Approach to Representation Theory
651
on the physics by working on concrete examples right at the start. Along the
same line, RepLAB can be used to quickly check whether an algebraic analysis
of the symmetries of a problem is worthwhile: while the bases returned are
approximate, the dimensions and multiplicities of the irreducible subrepresentations
are themselves not approximate.
Still, this manuscript presents an early version of our package, and it presents
several areas for improvement. Can we obtain better scaling factors for the algorithm
of Sect. 2? Currently, our running time is dominated by the eigendecomposition
step, which has running time similar to a single solver iteration on the original SDP.
Another route is to use the black-box algorithms above in the last resort, and exploit
structural information: RepLAB has built in support for a dozen standard group
and representation constructions, but is not exploiting that structure currently. Also,
RepLAB works in hardware double floating-point precision, and uses a tolerance of
ε = 10 −10 in all convergence tests. We are currently analyzing error propagation in
all the steps of our algorithm, and will replace those tolerances by proper bounds.
Finally, RepLAB performs limited exact solution recovery: can we extend those
recovery methods?
Acknowledgments 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).
References
1. A. Szanton, in The Recollections of Eugene P. Wigner, ed. by A. Szanton (Springer US, Boston,
MA, 1992), pp. 115–125. https://doi.org/10.1007/978-1-4899-6313-0_8
2. D. Rosset, J.D. Bancal, N. Gisin, J. Phys. A: Math. Theor. 47(42), 424022 (2014). https://doi.
org/10.1088/1751-8113/47/42/424022
3. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, S. Wehner, Rev. Mod. Phys. 86(2), 419
(2014). https://doi.org/10.1103/RevModPhys.86.419
4. C. ´
Sliwa, Phys. Lett. A 317(3–4), 165 (2003). https://doi.org/10.1016/S0375-9601(03)
01115-0
5. M. Mozrzymas, M. Horodecki, M. Studzi´ nski, J. Math. Phys. 55(3), 032202 (2014). https://
doi.org/10.1063/1.4869027
6. M. Studzi´ nski, M. Horodecki, M. Mozrzymas, J. Phys. A: Math. Theor. 46(39), 395303 (2013).
https://doi.org/10.1088/1751-8113/46/39/395303
7. M. Mozrzymas, M. Studzi´ nski, M. Horodecki, J. Phys. A: Math. Theor. 51(12), 125202 (2018).
https://doi.org/10.1088/1751-8121/aaad15
651
on the physics by working on concrete examples right at the start. Along the
same line, RepLAB can be used to quickly check whether an algebraic analysis
of the symmetries of a problem is worthwhile: while the bases returned are
approximate, the dimensions and multiplicities of the irreducible subrepresentations
are themselves not approximate.
Still, this manuscript presents an early version of our package, and it presents
several areas for improvement. Can we obtain better scaling factors for the algorithm
of Sect. 2? Currently, our running time is dominated by the eigendecomposition
step, which has running time similar to a single solver iteration on the original SDP.
Another route is to use the black-box algorithms above in the last resort, and exploit
structural information: RepLAB has built in support for a dozen standard group
and representation constructions, but is not exploiting that structure currently. Also,
RepLAB works in hardware double floating-point precision, and uses a tolerance of
ε = 10 −10 in all convergence tests. We are currently analyzing error propagation in
all the steps of our algorithm, and will replace those tolerances by proper bounds.
Finally, RepLAB performs limited exact solution recovery: can we extend those
recovery methods?
Acknowledgments 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).
References
1. A. Szanton, in The Recollections of Eugene P. Wigner, ed. by A. Szanton (Springer US, Boston,
MA, 1992), pp. 115–125. https://doi.org/10.1007/978-1-4899-6313-0_8
2. D. Rosset, J.D. Bancal, N. Gisin, J. Phys. A: Math. Theor. 47(42), 424022 (2014). https://doi.
org/10.1088/1751-8113/47/42/424022
3. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, S. Wehner, Rev. Mod. Phys. 86(2), 419
(2014). https://doi.org/10.1103/RevModPhys.86.419
4. C. ´
Sliwa, Phys. Lett. A 317(3–4), 165 (2003). https://doi.org/10.1016/S0375-9601(03)
01115-0
5. M. Mozrzymas, M. Horodecki, M. Studzi´ nski, J. Math. Phys. 55(3), 032202 (2014). https://
doi.org/10.1063/1.4869027
6. M. Studzi´ nski, M. Horodecki, M. Mozrzymas, J. Phys. A: Math. Theor. 46(39), 395303 (2013).
https://doi.org/10.1088/1751-8113/46/39/395303
7. M. Mozrzymas, M. Studzi´ nski, M. Horodecki, J. Phys. A: Math. Theor. 51(12), 125202 (2018).
https://doi.org/10.1088/1751-8121/aaad15