## Representation ring of SU(n)?

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What's the structure of representation ring of SU(n)?

What are the representations of generators?

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If $V$ is the standard representation, it's a free ring generated by $V$, $\wedge^2 V$, ..., $\wedge^{n-1}V$. This is very standard. – Will Sawin Nov 14 at 1:59

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I think the underlying question is fine (speaking as a non-expert) but that it would be better for you to provide some more motivation, some explanation of what you already know, etc. – Yemon Choi Nov 14 at 1:59

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@Will: to be fair it may be standard to many people but it might not be standard to everyone. My Mileage Does Vary. – Yemon Choi Nov 14 at 2:01

## 2 Answers

For a connected compact group, the representation ring is isomorphic to the subring of the representation ring of the maximal torus that is invariant under the action of the Weyl group.

The maximal torus of $SU(n)$ consists of the diagonal matrices in $SU(n)$. We can identify its representation ring with $Z[x_1,...,x_n]/(x_1...x_n-1)$, where $x_i$ is the representation that sends a diagonal matrix to the $i$th diagonal entry.

The Weyl group action of $S_n$ just permutes $(x_1,...,x_n)$. Thus, the representation ring is the ring of symmetric polynomials, generated by the elementary symmetric polynomials $s_1,...,s_n$, under the relation $s_n=1$.

It remains to show that the elementary symmetric polynomial $s_k$ corresponds to $\wedge^k$ of the standard representation. This is clear if we write the symmetric polynomial as a sum of monomials, and $\wedge^k$ of the standard representation of the maximal torus as a direct sum of $1$-dimensional representations. The decompositions are identical.

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Will has summarized concisely the classical structure theory for the representation ring of `$SU(n)$`, but it's worth emphasizing that all of this is found in textbooks on compact Lie groups. Doing any kind of research involving such older parts of representation theory requires some acquaintance with this kind of literature, to avoid re-inventing the wheel. It's also a good idea to place the special example in the context of simply connected semisimple compact Lie groups.

A typical source is the Springer GTM 98 Representations of Compact Lie Groups (1985) by Brocker and tom Dieck. They have a clear discussion of representation rings in section II.7, along with the general version of Will's answer in the setting of highest weights and fundamental representations in section VII.2. They also provide a lot of concrete details about classical groups, etc. (All of this material goes back to much older work of Weyl and others, and is treated in multiple sources.)

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This commentary on the literature is very welcome for interlopers like me who never learned any representations of Lie groups at "grad school level", beyond a sketchy look at SU(2), and consequently have to waste a lot of time poring through books in search of two or three pages that give what we need – Yemon Choi Nov 14 at 17:38
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