Phased permutation groups

[mkoeppe]

Mathematica uses "phased permutations" to express tensor symmetries.

A cycle of length k is labeled with a kth root of unity.

http://reference.wolframcloud.com/language/tutorial/TensorSymmetries.html

This generalizes the symmetries that sage.tensor can currently express, which are products of full symmetric groups (where the transpositions in the antisymmetries are labeled with -1).

We represent it as a matrix group in GL_n, and also provide a method that computes its representation as a subgroup of GL(T^{k,l)M).

Related reference: https://arxiv.org/pdf/2007.08056.pdf

CC: @tscrim @egourgoulhon @mjungmath @LBrunswic @mwageringel @dimpase @Ivo-Maffei

Component: combinatorics

Issue created by migration from https://trac.sagemath.org/ticket/30276

[mkoeppe]

Description changed:

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+++ 
@@ -9,4 +9,4 @@
 This generalizes the symmetries that `sage.tensor` can currently express, which are products of full symmetric groups (where the transpositions in the antisymmetries are labeled with -1).

-
+Related reference: https://arxiv.org/pdf/2007.08056.pdf

[mkoeppe]

Description changed:

--- 
+++ 
@@ -8,5 +8,8 @@

 This generalizes the symmetries that `sage.tensor` can currently express, which are products of full symmetric groups (where the transpositions in the antisymmetries are labeled with -1).

+We represent it as a matrix group in GL_n,
+and also provide a method that computes its representation as a subgroup of `GL(T^{k,l)M)`. 
+

 Related reference: https://arxiv.org/pdf/2007.08056.pdf

[dimpase]

comment:5

this seems to generalise to cyclic groups only, no?

[mkoeppe]

comment:6

each generator is a cycle...

[dimpase]

comment:7

each generator is a product of cycles, in full generality. Then, I think these things are called monomial groups, "phased" comes from physics people not taking algebra classes :-)

[mkoeppe]

comment:8

Yes, it's a Wolfram-ism, I think

[tscrim]

comment:9

Would these be a generalization of ColoredPermutations, where each element of {1, ..., n} can have its own distinct cycle length?

[mkoeppe]

comment:10

yes, but with some kind of compatibility relation, I guess.

[mkoeppe]

comment:12

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.