4
$\begingroup$

I have two questions:
It is well known that the complex representation ring $R(U(n))=\mathbb{Z}[\lambda_1,\cdots,\lambda_n,\lambda_n^{-1}]$, where $\lambda_1$ is the natural representation of $U(n)$ on $\mathbb{C}^n$ and $\lambda_i$ is the $i^{th}$ exterior power of $\lambda_1$. Is there any explicit description of the real representation ring $RO(U(n))$ in terms of generators and relations?
Lastly, is there a description for the adjoint representation $Ad(U(n))$ in terms of real representations?

Answers and/or sources would be much appreciated.

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ I think one needs at least $n(n+6)/8$ generators for $n$ even or $(n^2+ 8n-1)/8$ generators for $n$ odd, so one probably won't have a great presentation in terms of generators and relations. $\endgroup$
    – Will Sawin
    Commented Jul 16, 2022 at 15:38

0

Reset to default

You must log in to answer this question.