Questions tagged [frobenius-schur-indicator]

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4 votes
2 answers
704 views

If all real conjugacy classes are strongly real, then all real irreps are "strongly real"(symmetric), true?

Question Is true that if all real conjugacy classes of a finite group are strongly real, then all its real irreducible representations (irreps) are "strongly real" (symmetric)? And vice ...
Alexander Chervov's user avatar
19 votes
5 answers
1k views

Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?

Let $G$ be a simple algebraic group group over $\mathbb C$. Let $V$ be a self-dual representation of $G$. Let $\lambda$ be the highest weight of $V$. Write $\lambda$ as a sum of fundamental weights: $...
André Henriques's user avatar
6 votes
1 answer
324 views

A property forcing the Frobenius-Schur indicator to be positive

Let $G$ be a finite group. Two irreducible complex representations $V,V'$ of $G$ are called dual to each other if $V \otimes V'$ admits a trivial component, i.e. $\hom_G(V \otimes V',V_0)$ is positive ...
Sebastien Palcoux's user avatar
3 votes
2 answers
523 views

Frobenius-Schur indicator and character table of finite groups

Let $G$ be a finite group and $\pi$ an irreducible complex representation. The Frobenius-Schur indicator of $\pi$ is defined as: $$ \nu_2(\pi):=\frac{1}{|G|} \sum_{g \in G} \chi_{\pi}(g^2) $$ with $\...
Sebastien Palcoux's user avatar