# Representation ring of SU(n)?

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 '12 at 1:59
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 '12 at 1:59
@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 '12 at 2:01

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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Is it obvious that the generators you described satisfy no relations besides commutativity and associativity? –  Rasmus Bentmann Jul 8 '13 at 13:42
Yes, because of that fact on the ring of symmetric functions, which follows from counting symmetric functions of each degree. –  Will Sawin Jul 9 '13 at 6:35
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.