## The Category of Representations of a Group

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Do people study the category of representations of a compact finite group (not just irreducible ones)? I'm more interested in small cases like S_3 and SU(2) but I'd be curious about general cases like $S_n, SU(n)$. These must be tensor categories since - well... they admit tensor products and direct sums. Can these representations be considered a ring?

** What does this category look like in the case of S_3?

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Pick pretty much any textbook which deals with the representation of finite groups to find a description of the modules category of S_3... – Mariano Suárez-Alvarez Dec 30 2009 at 3:32

## 3 Answers

The category Rep(G) is a symmetric tensor category, and it is a theorem that this structure determines G (Tannaka-Krein duality, but I'm not familiar with it). Each object is dualizable because there is a dual representation, from which appropriate evaluation and coevaluation maps can be constructed. The unital object is the trivial representation.

(Incidentally, a symmetric tensor category is a kind of categorification of a commutative ring.)

In the finite case, It is a fusion category because there are finitely many simple objects, there is rigidity as stated above, and because $\mathbb{C}[G]$ (or more generally $k[G]$ for $k$ a field of characteristic prime to the order of $G$) is a semisimple algebra (Maschke's theorem). Semisimplicity is also true for continuous finite-dimensional representations of compact groups by the same "averaging" argument used in Maschke's theorem, though the group algebra is not necessarily semisimple. In the finite group case, the number of simple objects is equal to the number of conjugacy classes in G. In the infinite group case, for instance, the rotation group $SO(n)$ has infinitely many irreducible finite-dimensional representations obtained by the action on the spherical harmonics of various degrees (i.e. harmonic polynomials restricted to the sphere).

In the case of S_n, generators of this ring can be indexed by the Young diagrams of size n. The relations are given by the tensor product rules, and while the Pieri rule gives a special case of this, as far as I know, there is no a simple general way to express the tensor product of two representations associated to Young diagrams as a sum of Young diagrams.
However, there are apparently algorithms to do this.

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It's also worth mentioning that the dual and tensor structures "come from" the Hopf algebra structure of the group algebra, namely the antipode and comultiplication respectively. The same thing is true of the category of representations of any Hopf algebra. – Qiaochu Yuan Dec 30 2009 at 4:28

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In the usual statements of TK duality, the "category of representations" data consists of a (symmetric, etc.) category along with a faithful functor to VECT. See for example the paper "An introduction to Tannaka duality and quantum groups" by Ross and Street. – Theo Johnson-Freyd Jan 4 2010 at 4:36

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@Theo: Don't you mean Joyal and Street? (Ross is Street's first name.) You can find the paper here: maths.mq.edu.au/~street/CT90Como.pdf – José Figueroa-O'Farrill Feb 3 2010 at 23:50

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In what sense is $SU(n)$ a finite group? Perhaps your question is about compact (including finite) groups? In that case, the representation theory is very well understood. For compact groups, my favourite reference is the book by Bröcker and tom Dieck Represetations of compact Lie groups. Section II.7 describes the representation ring in general and introduces the Adams operations,... and then Section VI.5 contains the structure of the representation rings for the simply-connected classical Lie groups.

I should have added that the representations only form a semiring under direct sum and tensor product. The representation ring is obtained from this by the standard Grothendieck construction. You essentially have to add so-called virtual representations.

The book I mentioned does not treat the case of infinite-dimensional representations, though. Perhaps someone here can say more about this case.

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Do they not form a ring in the finite group case? – Grétar Amazeen Dec 30 2009 at 3:28
No, you don't have additive inverses.  You need to allow an irreducible representation to occur with negative multiplicity. – Qiaochu Yuan Dec 30 2009 at 3:40
Of course, thanks. I was thinking of formal combinations, but somehow wasn't :) – Grétar Amazeen Dec 30 2009 at 3:48

In the finite case at least, the answer is yes. It is a tensor category, and one fundamental result is that the category of representations of G is equivalent, as tensor categories, to the category of $\mathbb{C}G$-modules, where $\mathbb{C}G$ is the group algebra. The same holds if you just look at the finite dimensional representations, and finite dimensional modules if I remember correctly.

As for considering it as a ring, the answer is also yes, if you allow formal combinations of them (so called virtual representations) (thanks for pointing that out Jose and Qiaochu!). It has a basis the irreducible representations, and the multiplication is the tensor product, with the trivial representation as 1.

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What's your definition of the group algebra for an infinite compact group?  I've seen a few slightly different ones. – Qiaochu Yuan Dec 30 2009 at 3:09
I'm not sure, I'm only answering for the finite case.  – Grétar Amazeen Dec 30 2009 at 3:11
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