The symmetry group ${\rm Sym}(\Phi)$ of a frame $\Phi \in {\rm Mat}_{d \times n}(\mathbb{C})$ is a subgroup of the symmetric group $S_n$. An algorithm to compute ${\rm Sym}(\Phi)$ is described in Section 9.6 of Waldron's An Introduction to Finite Tight Frames.
We would like to implement Waldron's algorithm (which is appropriate for small symmetry groups) or another suitable algorithm to compute ${\rm Sym}(\Phi)$. It would be useful to compute any or all of the following.
-
${\rm Sym}(\Phi)$ as an explicit subgroup of $S_n$, described in terms of generators, compatible with the particular ordering of the vectors appearing in the database.
-
${\rm Sym}(\Phi)$ as an permutation group, that is, a direct product of one or more transitive permutation groups. Transitive permutation groups are catalogued in the LMFDB as Galois groups.
-
${\rm Sym}(\Phi)$ as an abstract group.
- The associated projective representation ${\rm Sym}(\Phi) \to {\rm PGL}_d(\mathbb{C})$.
- A genuine linear representation of an appropriate covering group ${\widetilde{{\rm Sym}(\Phi)} \to {\rm GL}_d(\mathbb{C}).}$
The symmetry group${\rm Sym}(\Phi)$ of a frame $\Phi \in {\rm Mat}_{d \times n}(\mathbb{C})$ is a subgroup of the symmetric group $S_n$ . An algorithm to compute ${\rm Sym}(\Phi)$ is described in Section 9.6 of Waldron's An Introduction to Finite Tight Frames.
We would like to implement Waldron's algorithm (which is appropriate for small symmetry groups) or another suitable algorithm to compute${\rm Sym}(\Phi)$ . It would be useful to compute any or all of the following.