Real representations of SO(n) and U(n)

Question

I would like to get some references where I can find the theory of the real representations of $\mathbf{SO}(n)$ and $\mathbf{U}(n)$.

In particular, I would like to know for which dimensions there exist irreducible representations of these groups and for which dimension, how many non-equivalent representations are?.

See Math Overflow: https://mathoverflow.net/questions/227869/is-there-a-formula-for-the-frobenius-schur-indicator-of-a-rep-of-a-lie-group — Jeffrey Adams, Apr 03 '20 at 03:07

score 5 · Answer 1 · answered Apr 01 '20 at 19:37

5

Take a look at John Frank Adams' Lectures on Lie groups. W. A. Benjamin, Inc., New York-Amsterdam 1969 xii+182 pp.. chapter 6.

answered Apr 01 '20 at 19:37

Nicolas Boerger

2,590

Bugs Bunny · Answer 2 · 2020-04-03T04:14:56.690

3

Read about Frobenius-Schur anywhere. In a nutshell a complex irreducible $V$ of complex dimension $n$ can give 1 or 2 real irreducibles, whose real dimensions are $n$ or $2n$. This can be easily determined by computing the FS-indicator or dimensions of invariants in both $S^2V$ and $\Lambda^2 V$. The latter can be done in Lie for each particular representation.

edited Apr 03 '20 at 04:14

answered Apr 02 '20 at 06:33

Bugs Bunny

12,113

3

Minor correction: $n$ or $2n$, not $n$ or $n/2$. – David E Speyer Apr 02 '20 at 14:55

score 3 · Answer 3 · answered Apr 03 '20 at 09:05

3

The book by Broecker and tom Dieck (Representations of Compact groups, Springer Graduate Texts in Math) has a very useful section about real, complex and quaternionic representations and how to pass between them.

answered Apr 03 '20 at 09:05

Vincent

2,437

Real representations of SO(n) and U(n)

3 Answers3