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 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.

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?

]]>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 |

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.

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

started *G-CW complex*.

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

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

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 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.

]]>have created an entry for *Bott periodicity*

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*

edited at *orbispace* in order to express Charles Rezk’s statement here more accurately.

the old entry *representation* contained an old query box with some discussion.

I am hereby moving this old discussion from there to here:

+– {: .query}
I don't agree with this $D \coloneqq Aut(V)$ business. A $k$-linear representation of a group $G$ is a functor from $\mathbf{B}G$ to $k Vect$, period. Because $\mathbf{B}G$ has one object (or is pointed), we can pick out an object $V$ of $k Vect$, and it was remiss of me not to mention this (and the language ‘*on* $V$’ vs ‘*in* $D$’. But we usually don't want $D$ to actually *be* $Aut(V)$ instead of $k Vect$; when doing representation theory, we fix $G$ and fix $k$ (or fix $D$ in some other way), but we don't fix $V$. —Toby

If you look at the textbooks of representation of groups, then they start with representation of groups as homomorphisms of groups, that is just functors. Then they say, that usually the target groups are groups of automorphisms of some other objects. And at the end they say that one usually restricts just to linear automorphisms of linear objects when linearizing the general problematics to the linear one. Now the fact that in some special case there is a category which expresses the same fact does not extend to other symmetry objects, like for representations of vertex operator algebras, pseudotensor categories etc. I mean End(something) or Aut(something) is just inner end in some setup like in closed monoidal category, but there are symmetries in mathematics which have a notion of End of Aut for a single object but do not have good notion of category one level up which has inner homs leading to the same End or Aut. Conceptually actions are about endosymmetries or symmetries (automorphisms) being reducable to categorical ones but not necessarily, I think. In a way you say that you are sure that any symmetry of another object can be expressed internally in some sort of a higher category of such objects, what is to large extent true, but I am sure not for absolutely all examples.

- I can’t recall ever seeing group homomorphisms $\rho\colon G \to H$ described
*in general*as ’representations’, but I have limited experience; I should look at some more textbooks. The one that I learnt the subject from, Serre's*Linear Representations of Finite Groups*, looked only at representations on vector spaces from the beginning, but its title suggests a bias that might explain that. (^_^)

(for “on” terminology:) Ross Street uses monads in a 2-category and monads on a 1-category and I know of no objects in category theory.

- Yes, this is analogous to representation
*in*a category vs*on*an object in such a category. (But what do you mean by ’I know of no objects in category theory’?)

Another important thing is that the endomorphisms are by definitions often equipped with some additional (e.g. topological) structure which is not necessarily coming from some enrichement of the category of objects. –Zoran

- Good point.

(Zoran on word “classical representation” being just for groups: so the representations of associative algebras, Lie algebras, Leibniz algebras, topological groups, quivers, are not classical ??).

- I thought that they came later, but maybe not. I added ’of groups’ to fix/clarify. —Toby =–

In order to formalize some physics, I am looking for a suitable mathematical concept. It looks like the putative concept ought to be something like “pro-finite $\mathfrak{su}(2)$-representations”, but I am not sure yet. And once I am sure, I’ll be wondering if there is any decent established theory for such things.

The following is the motivation (taken from here):

**Motivation**

There is the remarkable observation (MSJVR 02, checked in AIST 17) that in the BMN matrix model supersymmetric M2-M5-brane bound states are identified with “limit sequences” of isomorphism classes of finite-dimensional complex Lie algebra representations of su(2).

Concretely, if

$\mathbf{N}( \{N^{(M2)}_i,N^{(M5)}_i\}_i ) \;\;\coloneqq\;\; \underset{ i }{\oplus} N^{(M2)}_i \cdot \rho_{N^{(M5)}_i} \;\;\in\;\; \mathfrak{su}(2)Rep^{fin}$denotes the representation containing

- $N^{(M2)}_i$ direct summands

of the

- $N^{(M5)}_i$-dimensional irrep

(for $\{N^{(M2)}_i, N^{(M5)}_i\}_{i} \in (\mathbb{N} \times \mathbb{N})^I$ some finitely indexed set of pairs of natural numbers)

with total dimension

$N \;\coloneqq\; dim\big( \mathbf{N}( \{N^{(M2)}_i,N^{(M5)}_i\}_i ) \big)$then:

an M5-brane configuration corresponds to a sequence of such representations for which

$N^{(M2)}_i \to \infty$

for fixed $N^{(M5)}_i$

and fixed ratios $N^{(M2)}_i/N$

an M2-brane configuration corresponds to a sequence of such representations for which

$N^{(M5)}_i \to \infty$

for fixed $N^{(M2)}_i$

and fixed ratios $N^{(M5)}_i/N$

for all $i \in I$.

Hence, by extension, any other sequence of finite-dimensional $\mathfrak{su}(2)$-representations is a kind of mixture of these two cases, interpreted as an M2-M5 brane bound state of sorts.

$\,$

**Question.** I’d like to extract a precise definition of “M2-M5 brane bound state” from the above. It must subsume suitable limits of finite-dimensional $\mathfrak{su}(2)$-representation as above. But taken where? And identified how?

Is “profinite $\mathfrak{su}(2)$-reps” a thing in representation theory? (i.e. pro-objects in the category of finite-dimensional representations.) Or maybe ind-objects instead? Or something else?

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