added to *S-matrix* a useful historical comment by Ron Maimon (see there for citation)

added to *E7* the statement of the decomposition of the smallest fundamental rep under $SL(8,\mathbb{R})$ and $SL(7,\mathbb{R})$ (here) and used this then to expand the existing paragraph on *As U-duality group of 4d SuGra*

I am touching various entries related to equivariant stable homotopy theory, adding basics from the literature. For instance I briefly added to *G-spectrum* the basic definition via indexing on a universe, and added the statement of the equivariant stable Whitehead theorem, cross-linked with the relevant bits at *equivariant homotopy theory*, etc. I have also been expanding a little more at *RO(G)-grading* and cross-linked more with old material at *equivariant cohomology*. Tried to make the link between RO(G)-grading and equivariant suspension isomorphism more explicit.

Just in case you are watching the logs and are wondering. I am not announcing every single edit, unless there is anything noteworthy.

]]>I have tried to expand the Idea-section at *orbit method* a little.

have created an entry for *Bott periodicity*

added to *quiver* a very brief remark on the *Gabriel classification theorem*

Started a bare minimum at *cyclotomic spectrum*. So far it’s essentially just a pointer to the canonical reference by Blumberg-Mandell. (Thomas Nikolaus and Peter Scholze have a new foundation of the theory in preparation for which notes however are not public yet, also Clark Barwick has something in preparation, for which you may find notes by looking at his website and being clever in deducing hidden URLs, he says.)

For the moment the only fact that I have actually recorded in the entry is a fact that is trivial for anyone familiar with the theory,but which looks interesting from the point of view of the story at *Generalized cohomology of M2/M5-branes (schreiber)*: the global equivariant sphere spectrum for all the cyclic groups (all the A-type finite groups in the ADE classification…) carries canonical cyclotomic structure and as such is the tensor unit among cyclotomic spectra.

Apart from mentioning this, I have added brief cross-links with *topological cyclic homology*, *equivariant sphere spectrum*, *cyclic group* and maybe other entries.

gave *representation theory* a little Idea-section, then added some words on its incarnation as homotopy type theory in context/in the slice over $\mathbf{B}G$ and added the following *homotopy type representation theory – table*, which I am also including in other relevant entries:

homotopy type theory | representation theory |
---|---|

pointed connected context $\mathbf{B}G$ | ∞-group $G$ |

dependent type | ∞-action/∞-representation |

dependent sum along $\mathbf{B}G \to \ast$ | coinvariants/homotopy quotient |

context extension along $\mathbf{B}G \to \ast$ | trivial representation |

dependent product along $\mathbf{B}G \to \ast$ | homotopy invariants/∞-group cohomology |

dependent sum along $\mathbf{B}G \to \mathbf{B}H$ | induced representation |

context extension along $\mathbf{B}G \to \mathbf{B}H$ | |

dependent product along $\mathbf{B}G \to \mathbf{B}H$ | coinduced representation |

made *completely reducible object* redirect to *semisimple object*.

(Also touched the formatting at *simple object*).

I am slowly creating a bunch of entries on basic concepts of equivariant stable homotopy theory, such as

- equivariant suspension spectrum, equivariant sphere spectrum, equivariant homotopy groups, RO(G)-grading, fixed point spectrum, tom Dieck splitting

At the moment I am mostly just indexing Stefan Schwede’s

]]>started *Elmendorf’s theorem* with a brief statement of the theorem

have noted down the basic properties of the irreducible representations of the Lorentzian spin group, at *spin representation – Properties*.

started some minimum at *MR cohomology theory*

wrote an entry *Deligne’s theorem on tensor categories* on the statement that every regular tensor category is equivalent to representations of a supergroup. Added brief paragraphs pointing to this to *superalgebra* and *supersymmetry*, added cross-links to *Tannaka duality*, *Doplicher-Roberts reconstruction* etc. Also created a disambiguation page *Deligne’s theorem*

started something at *ADE classification*, but am out of steam (and time) now.

I have split off *group character* from *character* and added discussion of how character groups of tori are isomorphic to their fundamental group.

What happens to this statement for more general ellitpic curves?

]]>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.

I have started *rational equivariant stable homotopy theory*, but so far there is nothing but references.

added to *equivariant K-theory* comments on the relation to the operator K-theory of crossed product algebras and to the ordinary K-theory of homotopy quotient spaces (Borel constructions). Also added a bunch of references.

(Also finally added references to Green and Julg at *Green-Julg theorem*).

This all deserves to be prettified further, but I have to quit now.

]]>started *G-CW complex*.

stub for *Atiyah-Segal completion theorem*, for the moment just to record a reference

gave a bare minimum to *Schur orthogonality relation* (for the moment just because I want to be able to link to it).

I needed a table *exceptional spinors and division algebras – table*, and so I have created one and included it into the relevant entries

I have created an entry *modular equivariant elliptic cohomology*.

The subject barely exists, for the moment the entry is to serve two purposes:

first, to highlight that by results of Mahowald-Rezk, Lawson-Naumann, Hill-Lawson this exists as a rather compelling generalization of KR-theory;

second, that the close the relation of KR-theory to type II string theory on orientifolds has previously been conjectured to correspond in the lift of the latter to F-theory to a modular equivariant universal elliptic cohomology.

So while the subject hasn’t been studied yet (it seems), both its construction and plenty of motivation for it already exists. And now also an $n$Lab entry for it does. :-)

]]>have split-off a stub *positive energy representation* from *loop group*