# Questions tagged [characters]

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

650 questions
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### Additive characters of $\mathbb{C}_p$

Consider $x \in \mathbb{C}_p$ with $|x|<1$ then for $a \in \mathbb{Z}_p$ we have the characters $$a \mapsto (1+x)^a$$ where $(1+x)^a= \exp(a\log_p(1+x))$ My question is : it's possible to ...
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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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### Jacobi sums Gaussian Sum. Show $J(\rho^{'},\chi^{'}) = (- J(\rho,\chi))^s$

I want to show that $J(\rho^{'},\chi^{'}) = (- J(\rho,\chi))^s$, where $\rho^{'},\chi^{'}$ are characters of a finite field $F_{p^s}$ and $\chi,\rho$ are characters a finite field $F_p$. My work: I ...
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### orthogonality relation for characters

Let $\rho : G \to GL(V)$ be a representation on G. Then, its character is defined as $\chi_V(g) := Tr(\rho(g))$, where $Tr$ denotes the trace function. For an exercise I am trying to solve, I would ...