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I noticed that some time ago Domenico had made some notes at projective representation on the homotopy-theoretic / 2-groupoid-theoretic formulation. I have now expanded this discussion a good bit, here.
For when editing functionality is back:
While fairly straightforward, it’s useful to make explicit that character theory by and large still works for projective representations:
And while I am at it, some general background references:
Jürgen Tappe, Irreducible projective representations of finite groups, Manuscripta Math 22, 33–45 (1977) (doi:10.1007/BF01182065)
Tania-Luminiţa Costache, On irreducible projective representations of finite groups, Surveys in Mathematics and its Applications 4 (2009), 191-214 (ISSN:1842-6298)
Eduardo Monteiro Mendonça, Projective representations of groups, 2017 (pdf)
It occurs to me that what is known as “the non-commutative torus” may be thought of as the characterization of projective intertwiners of the regular $\mathbb{Z}/n$-representation regarded as a projective representation.
Not a big deal, but is this perspective expanded on anywhere? What I mean is this:
For $n \in \mathbb{N}$ regard the regular $\mathbb{Z}/n$-representation matrix
$\big( U_{reg}([k]) \big)_{q q'} \,=\, e^{ 2 \pi \mathrm{i} k \cdot (q-1) / n } \cdot \delta_{q q'}$as a projective representation
$U_{reg} \;\colon\; \mathbb{Z}/n \xrightarrow{ \; } \mathrm{U}(n) \xrightarrow{\;} PU(n)$and then ask for a projective self-intertwiner, hence for
$U_2 \;\in\; U(n) \to PU(n)$such that
$\underset{k}{\forall} \;\;\; U_{reg}([k]) \cdot U_2 \;=\; U_2 \cdot U_{reg}([k]) \;\;\; \in \; PU(n)$hence such that
$U_{reg}([1]) \cdot U_2 \;=\; U_2 \cdot U_{reg}([1]) \;\;\; \in \; PU(n)$hence such that
$U_{reg}([1]) \cdot U_2 \;=\; \theta \cdot U_2 \cdot U_{reg}([1]) \;\;\; \in \; \mathrm{U}(n)$for some $\theta \,\in\, \mathrm{U}(1) \xhookrightarrow{diag} U(n)$.
This is the defining equation for the “non-commutative torus” (e.g. (2.30) on p. 10 of arXiv:hep-th/9912130, just to point to the next best reference that Google digs out), whose existence hence equivalently means that such a projective intertwiner $U_2$ exists, given by the one-step cyclic permutation matrix. Generally, such projective intertwiners are given by all the cyclic permutation matrices, I suppose.
This is, I think, one way to see concretely that the space of “stable” maps $\mathbf{B} (\mathbb{Z}/n) \xrightarrow{\;} \mathbf{B} PU(\mathcal{H})$ is $Map\big( B (\mathbb{Z}/n),\, B^3 \mathbb{Z} \big) \;\simeq\; B Hom\big(\mathbb{Z}/n, \, \mathrm{U}(1)\big) \times B^3 \mathbb{Z}$.
Maybe this relation to the NC torus has no significance, it just occurred to me as being somewhat curious. One subtlety here is that much of the literature considers projective representations with non-projective intertwiners between them (e.g. bottom of p. 2 in #2 above), in which case Schur’s lemma holds (e.g. Lemma 2.1 in #2 above) which would rule out the above NC torus-type example.
More generally, for $\mathbb{Z}/n$ replaced by any finite group $G$, it should be the case that every group character $\kappa\in Hom\big(G, U(1)\big)$ induces a projective endo-twiner of the regular $G$-rep, given on the irrep labels by tensoring with the group character. Now the phase $\theta = \theta(g)$ above is given by the very values of that group character.
For finite groups with vanishing $H^3(G; \mathbb{Z})$ this construction gives a concrete realization of how the space of “stable” morphisms $\mathbf{B}G \to \mathbf{B} PU(\mathcal{H})$ is $Map(B G, B^3 \mathbb{Z}) \simeq B Hom\big(G, U(1) \big) \times B^3 \mathbb{Z}$.
Hi Harry, we don’t currently edit pages, as we are waiting for Richard Williamson to turn the edit functionality back on. See the thread nLab migration to the cloud.
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