# Tagged Questions

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### Parabolic characters of subgroups $\Gamma \subset \operatorname{SL}_2(\textbf{Z})$ generated by parabolic and elliptic elements [on hold]

In the paper Generalized Modular Forms from Knopp and Mason, one can read in page $6$: Remark. It is not too hard to prove that a subgroup $\Gamma$ of finite index in $\operatorname{SL}_2(\textbf{Z})$...
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### Relation of these two Dirichlet $L$-functions

Let $\chi$ and $\psi$ be two quadratic Dirichlet characters and let $L(s,\chi)$ and $L(s,\psi)$ their associated Dirichlet $L$-functions. Is there a realtion between these two Dirichlet $L$-functions:...
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### Dirichlet Characters as Eigenvectors

This was asked in Math Stackexchange here but generated no comments or answers. I have slightly edited the original question with the comment in the fourth paragraph and the explicit matrix example at ...
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### Quadratic Gauss sums: Explicit determinations?

Can anyone please tell me (give me a reference, preferably) if there is any explicit determination of sums of the form $g(n,\chi):=\sum_{r=1}^{q}\chi(r)e(\frac{rn+r^2}{q})$ where $\chi$ is a Dirichlet ...
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### Extension of a formula for the quadratic Gauss sums

I am interested in the sums $$g(a,k)=\sum_{n=0}^{p-1}e^{2\pi i a n^k/p}$$ where $p\equiv1\mod k$ is a prime and $a$ is coprime with $p$. When $k=2$, it is a classical fact that $g(a,2)=\chi(a)g(1,2)$ ...
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### Asymptotic Weyl Character Formula

Let $G$ be a complex semi-simple group along with a chosen pair of opposite Borel subgroups (so we get all the root-theoretic data we need). Let $\lambda$ be a dominant weight, and let $V(\lambda)$ be ...
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### Missing formula! [closed]

I am doing a project on group association schemes, in particular looking at the structure constant $$p_{KL}^M = \#\{(x, y, xy) : x \in K, y\in L, xy \in M\}$$ where $K, L$ and $M$ are conjugacy ...
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### Hecke Character vs Grossencharakter

I would like to know if there is any difference between (1) an algebraic Hecke character (2) a Hecke character (3) a Grössencharakter All of the above in the setting of ellitpic curves with complex ...
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### On Applications of Murnaghan Nakayama Rule

This question is crossposted at math.stackexchange here and may be beyond the usual scope of the site. The question is located below. In short, I am looking for an accessible explanation of the ...
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### How much of character theory can be done without Schur's lemma or the Artin-Wedderburn theorem?

This is a somewhat imprecise question, as I am not sure how exactly how to formalise how to do mathematics "without" a certain key tool, but hopefully the intent of the question will still be clear. ...
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### On finite groups with same complex-valued character table

What are the necessary and sufficient conditions for two finite groups $G$ and $H$ to have same complex-valued character table? Is there any criterion for which one could know about the character ...
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### Characters of p-groups

Berkovich mentioned the following result of Mann in his book on p-groups: The number of nonlinear irreducible characters of given degree in a p-group is divided by p-1. Do you know any reference for ...
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### Character table entries and sums of roots of unity

It is well-known that the entries of the character table of a finite group are sums of roots of unity. Question: Is the converse true? Explicitly, given $z\in \mathbb{Z}[\mu_\infty]$, can I find a ...