Questions tagged [frobenius-schur-indicator]

Filter by
Sorted by
Tagged with
4
votes
1answer
160 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 ...
2
votes
0answers
182 views

Finite simple groups and negative Frobenius-Schur indicator

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
votes
2answers
202 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 $\...
4
votes
2answers
312 views

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 ...
5
votes
0answers
105 views

A class function defined using Frobenius-Schur indicators

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
votes
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 ...
17
votes
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: $...
2
votes
2answers
551 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 versa? ...
1
vote
1answer
327 views

sum_g g^k, Frobenius-Schur indicators, S_n-invariants in freeAss(x_i), center of the group algebra

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 $...