# Questions tagged [characters]

For questions about characters (traces of representations of a group on a vector space).

668 questions
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### Show that the characters of the representations $\phi_{n}$ of $SU(2)$ constitute a complete orthogonal set.

The question is given below: And the other questions mentioned are (I know the solutions of all of them): Sorry for the bad formulation of the my question at the first time I have ...
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### Isomorphic representations have equal characters

I have read the following sentence in Vinberg "linear representations of groups" pg.58: "Since isomorphic representations are given in compatible basis by identical matrices, we see that isomorphic ...
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### Problem 3.8 from Martin Isaacs' Character theory of Finite groups.

Let $\chi$ be a (possible reducible) character of G which is constant on G-{1}. Show that $\chi=a1_{G}+b\rho_{G}$ where a,b $\in \mathbb{Z}$ and $\rho$ is a regular character of G. Also show that if ...
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### Calculate the characters of the left and right regular representationsof an arbitrary finite group.

Calculate the characters of the left and right regular representations of an arbitrary finite group. The answer of the question is given below: But I do not know why the character of the left and ...
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### Convolution of trig polynomials over a Group

I want to prove that $T(G)=T(G)*T(G)$ where G is an infinite compact abelian Hausforff Topological group. I'm trying to start this but really im confused with the convolution. Say $f,g \in T(G)$ I ...
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### Solution Verification: Irreducible representations of $C_n$ present in $V \otimes V$.

Let $C_n$ be the cyclic group order n. Let V be the faithful two dimensional representation (over complex field) denoted by: \rho(g^j)= \begin{pmatrix} \omega^j & 0 \\ 0 &...
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### Proof Verification: Let $f(\chi)$ be the conductor of $\chi$, proof that $f(\chi)=f(\chi_1)\cdots f(\chi_r)$.

This is a detailed problem, let me write down the problem and the process I have done: Assume that $k=k_1k_2\cdots k_r$ where $k_i$ and $k_j$ are relatively prime for $i\neq j$. Let $\chi$ be a ...
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### 6-Dimensional Irrep of $S_5$

So I am computing the character table for $S_5$, and the only thing I have yet to understand is how we know the character values for the row relating to the 6-dimensional irrep. The irreps I have are: ...
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### understanding the irreducible Brauer characters of a defect-1-block

In section XII of his famous paper from 1966, Janko investigated the principal 11-block of his group $J_1$ (and thereby finally proved existence and uniqueness of this group). I would like to learn ...
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### Group characters are eigenfunctions of the Laplacian with eigenvalue proportional to the quadratic Casimir

For the finite dimensional irreducible representations of $SU(2)$ we have that the group characters $\chi_n(g)$ for the $n^{th}$ representation are eigenfunctions of the Laplacian over the group ...
### Embedding $\mathbb C_p$ into $\mathbb C$ and vice versa…?
I'm reading Washington's book on cyclotomic fields, and he mentions that it is sometimes convenient to embed $\mathbb{C}_p$ into $\mathbb C$ and vice versa. In my mind, $\mathbb C_p$ and $\mathbb C$ ...