# Questions tagged [frobenius-schur-indicator]

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### 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 ...
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### Is $1\neq a\in Z(2.E_7(q))\cong Z_2$ a square element in $2.E_7(q)$?

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 ...
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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 $\epsilon(\chi):=\frac{1}{|G|}\sum_{g\in ... 0answers 64 views ### Interpretation of Hilbert/Frobenius series shift Let$V = \oplus_{i\geq 0} V^i$be a graded vector space. Recall that the Hilber series is defined as $$F(q) = \sum_{i\geq 0} q^i dim(V^i),$$ or if we have a graded$S_n$-module,$M$, we can ... 5answers 932 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:$...
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 \$...