Questions tagged [characters]
For questions about characters (traces of representations of a group on a vector space).
1,140
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Serre's Representation Theory exercise 7.3(d)
I am trying to solve Exercise 7.3(d) in Serre's Linear Representation of Finite Groups. I have solved all other parts. The important points of where I am stuck at boils down to the following facts. (I ...
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1
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Does the formal character determine the representation?
Suppose $V,W$ are two finite-dimensional representations of a Lie algebra $\mathfrak{g}$.
Is it true that if their formal characters coincide, $$\mathrm{ch}_V=\mathrm{ch}_W ,$$ then the ...
- 1,562
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On Frobenius–Schur indicator of real/complex representations
Let $G$ be a finite group with complex irreps $W_i$. Let $V$ be a real irrep of $G$. Denote $\chi_{W_i}$ and $\chi_{V}$ the corresponding characters.
Each $V$ has three possibilities:
Case 1: $\dim_{\...
- 2,307
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Set of primitive idempotents of group algebras
Let $G$ be a finite group and $K$ be a field with characteristic zero. Can we construct the set of primitive orthogonal idempotents of $KG$? By this set, I mean the set of idempotents such that $...
- 2,307
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Can the sum of a nonlinear irreducible character's values on $Z(\chi)$ be zero? [closed]
I need a lemma for a research problem. Suppose that I sum the values of a nonlinear irreducible character $\chi$ of a finite group over the center of that character $Z(\chi)$. Is it possible for the ...
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Brauer characters/ modular characters are not well defined
In chapter $15$ of Isaacs' "Character theory of finite groups", he defines a field $F$ of characteristic $p$, isomorphic to the algebraic closure $\overline{\mathbb{F}_p}$ of the prime field ...
- 115
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Character of a kac-moody module as the sum of characters of Verma modules
I'm trying to prove the next result from Kac's book, Infinite dimensional Lie algebras.
Let $V$ be a $\mathfrak{g}(A)$-module with highest weight $\Lambda$. Then
$$
\text{ch}(V)=\sum_{\lambda\in B(\...
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What are the irreducible representations of $A_3$?
I've got some notes saying the Character table of the alternating group $A_3$ is given as in the attached image. I can't seem to figure out what the representations $\rho_1, \rho_2$ are supposed to be....
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Decomposition of primitive central idempotents in group algebras
Let $W$ be an irreducible $\mathbb{C}$-representation of a finite group $G$ with character $\chi_W$. A primitive central idempotents of the group algebra $CG$ is: $$e=\frac{\dim_{\mathbb{C}}(W)}{|G|}\...
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Exercise 2.19 in Isaacs's book on character theory
Here is the problem:
Let $E=\langle x_1,x_2,x_3,x_4\rangle$ be an elementary abelian group of order 16.Let $P=\langle y\rangle$ be cyclic of order 3.$P$ acts on $E$ by
$$x_1^y=x_2,x_2^y=x_1x_2,x_3^y=...
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Problem 2.14 from Isaacs's Character Theory of Finite Group
I'm solving this problem from Isaacs's Character Theory of Finite Group:
Let $H \subseteq G' \cap Z(G)$ be cyclic of order $n$ and let $m$ be the maximum of the orders of the elements of $G/H$. ...
- 3
1
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1
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How to determine $\Gamma_{15}$?
For context,yesterday I asked How to determine $\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4,\Gamma_5$?.
Now I learned of new theorem that made me curious:
Theorem:
Let $m,n \in \mathbb{Z}$ such that $gcd(m,n)=...
- 609
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Non-commutative banach algebra
A theorem states that if $A$ is a commutative Banach algebra, then for all $a\in A$
we have
$$\sigma(a)=\{\chi(a) : \chi \text{ is a character }\}$$
My question what if $A$ is not commutative. Does ...
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Generic bound on quadratic character sum
Let $\chi$ be the non-trivial quadratic character of $\mathbb{F}_q$, and let $f(x)$ be a square-free polynomial over $\mathbb{F}_q[x]$. Then by the Weil bound, we have the generic estimate $|\sum_{x\...
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Dimension of the center of a block
Let $G$ be a finite group and let $F$ be an algebraically closed field of characteristic $p$. I've been studying some modular character theory from Navarro's "Characters and blocks of finite ...
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