# Questions tagged [frobenius-schur-indicator]

The frobenius-schur-indicator tag has no usage guidance.

The frobenius-schur-indicator tag has no usage guidance.

11
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EDIT (November 1, 2022): Over the weekend I think I found a technique to determine the exact multiplicities, according to how conjugation acts on the fundamental weights. While I haven't done the ...

1
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0
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A near-group category $G+m$ is a (spherical) fusion category whose simple objects are given by the element $g$ of the finite group $G$, plus one extra simple object $y$, with Grothendieck ring as ...

6
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1
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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 ...

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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 $\...

3
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2
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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 $\...

4
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2
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When $q$ is a power of some odd prime, is $1\neq a\in Z(2.E_7(q))\cong Z_2$ a square element in $2.E_7(q)$?
A Lie algebra is a vector space $L$ over a field $K$ on which a product operation $[xy]$ is ...

7
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1
answer

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Let ${\rm Irr}(G)$ be the set of complex irreducible characters of a finite group $G$. The Frobenius-Schur indicator of $\chi\in{\rm Irr}(G)$ is defined to be $\epsilon(\chi):=\frac{1}{|G|}\sum_{g\in ...

5
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0
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Let $V = \oplus_{i\geq 0} V^i$ be a graded vector space.
Recall that the Hilbert series is defined as
$$F(q) = \sum_{i\geq 0} q^i \operatorname{dim}(V^i),$$
or if we have a graded $S_n$-module, $M$, ...

18
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5
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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: $...

2
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2
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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 versa?
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For any $k$ sums $T_k = 1/|G|\sum_{g\in G} g^k$ belong to the center of the group algebra, for finite group G.
For $k=2$ they "are" (up to details and interpretation) Frobenius-Schur indicators.
For $...