self-dual representations

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

Let $V$ be a finite-dimensional irreducible representation (complex or $\ell$-adic) of a group $G$ (compact Lie group or algebraic group etc.). Does there always exist a linear character $\rho$ of $G$, such that $V\otimes\rho$ is a self-dual irrep. of $G?$ Namely $V\otimes\rho\simeq(V\otimes\rho)^*.$ If not, is there any necessary/sufficient conditions on $V$ for it to be "twisted self-dual"?

If this is always the case, then in particular, if $G$ has no non-trivial linear characters (e.g. $G$ is a simply-connected compact Lie group or a perfect finite group), then every irrep. of $G$ is self-dual.

Thanks.

-

1
The question isn't formulated precisely enough at this point to be answered directly. But in any case the complex irreducible representations of a simply connected compact Lie group certainly aren't always self-dual as one sees immediately in rank 2 examples. – Jim Humphreys Jun 12 2011 at 22:33
Isn't the natural representation of $\mathrm{SL}(n)$ a counter example to the first question? – Bruce Westbury Jun 12 2011 at 22:34
@Bruce: Yes, or similarly for compact groups such as `$SU(n)$` in the context of the question and my comment. – Jim Humphreys Jun 12 2011 at 23:35
This already fails for finite groups. For instance, $PSL(2,7)$ has no nontrivial linear characters, but has two non-isomorphic 3-dimensional irreducible reps which are dual to each other. – Kevin Ventullo Jun 13 2011 at 6:48

1 Answer

If you want a representation $V$ to be self-dual up to a character, then either $S^2V$ or $\Lambda^2V$ (considered as representations of $G$) should have a 1-dimensionanal summand (corresponding to the isomorphism $V \to V^*\otimes\chi$). But as it was mentioned by Jim there are a lot of representations $V$ for which both $S^2V$ and $\Lambda^2V$ are irreducible. For example, this is the case for $G = SL(n)$ and $V$ the standard representation --- the example suggested by Bruce.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 24, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9215624332427979, "perplexity_flag": "head"}