Questions tagged [characters]
For questions about the algebraic concept of 'character': a function from a group into a field satisfying certain properties. Not to be confused with the more commonly known psychological term.
268 questions
3
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Averages of exponential-character sums over primes
Let $$\psi _{\chi }(\beta ):=\sum _{n\leq x}\Lambda (n)\chi (n)e(n\beta ).$$ A result like $$\sum _{1<q\leq Q}\sideset {}{^*}\sum _{\chi \text { mod }(q)}|\psi _\chi (0)|\ll Q^2\sqrt x$$ is ...
- 1,706
-2
votes
0
answers
52
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On a partial sum with Moebius function and Dirichlet characters
Considering a Dirichlet Character $\chi(n)$, I would like to study the convergence of following sums in the critical strip:
$S=\sum_{n=1}^{\infty} \sum_{r|n} \mu(\frac{n}{r}) \chi(r) r^{-s}$
Any ...
1
vote
0
answers
76
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Why is the Goss plane defined as it is on 1-units?
I am reading the preprint by Kramer-Miller and Upton on zeros of the Goss zeta function, and I am wondering why the Goss plane is defined the way it is.
For $K$ a global function field of ...
- 111
2
votes
0
answers
75
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A character isometry in $A_5$
Background:
Let $G$ be a finite group. Fix a prime $p$. We say that $H\subseteq G$ is strongly $p$-embedded if:
$p$ divides $|H|$;
For every $x\in G-H$, $p$ does not divide $|H\cap H^x|$.
Some facts ...
- 2,413
6
votes
0
answers
195
views
Character and exponential sums over primes under GRH
Theorem 18.13 of Montgomery and Vaughan's multiplicative number theory book says for non-principal characters (ignoring $\log $'s) $$\int _{-\delta }^{\delta }|\psi _\chi (\beta )|^2d\beta \ll \delta ...
- 1,706
6
votes
2
answers
172
views
Dimension of the $H$ fixed subspace of tensor product of representations
Let $G$ be a finite group and $H$ be a normal subgroup of $G$ of index $2$. Let $\operatorname{IRR}(G)$ denote the set of all inequivalent irreducible representations of $G$. For any representation $(\...
- 91
2
votes
1
answer
262
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Eigenvalues of a sum of tensor product of representation matrices
Let $G$ be a finite group and $H\le G$. Let $\lbrace{\rho_1,\rho_2,\ldots,\rho_t}\rbrace$ be the set of inequivalent, irreducible representations of $G$, where $\rho_1$ is the trivial representation ...
- 123
0
votes
0
answers
126
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Describing imaginary abelian fields in terms of Dirichlet characters
I am reading the paper, titled "The imaginary abelian number fields with class numbers equal to their genus class numbers" in which the authors describe an imaginary abelian field $N$ by the ...
- 597
5
votes
1
answer
188
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Reference request: $p$-local Frobenius complements in finite groups
Let $G$ be a finite group, and let $p$ be a prime.
Let $H\subseteq G$ be a subgroup, where $p$ divides $|H|$. We shall say that $H$ is a $p$-local Frobenius complement if $H\cap H^x$ is a $p'$-group ...
- 2,413
7
votes
1
answer
269
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Pullback of characters from $\operatorname{SO}(n^2)$ to $\operatorname{SO}(n)$ via the tensor representation
$\DeclareMathOperator\SO{SO}$I am trying to prove that the Haar measure over $\SO(n^2)$ is "close enough" (probably in some Wasserstein sense) to the pushforward of the Haar measure on $\SO(...
- 103
3
votes
1
answer
370
views
Brauer induction with negative coefficients
Let $G$ be a finite group. The Brauer induction theorem says that every irreducible complex character of $G$ is an integer linear combination of characters induced from degree $1$ characters of ...
- 613
3
votes
1
answer
350
views
Gauss sum and p-adic Gamma function
Let $p$ be a prime and and for a character $\chi$ of $\mathbb{F}_p^\times$, we define $\bar{\chi}(a)=\chi^{-1}(a)$.
Let $\chi$ has order $k$ such that $\bar{\chi}$ is the power residue symbol modulo $...
- 53
3
votes
1
answer
193
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Existence of Irreducible representation of finite group with central kernel
There are many nice results on when a finite group $G$ has a faithful complex irreducible representation. For example a finite nilpotent group has such representation if and only if its center $\...
7
votes
0
answers
241
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Irreducible representation with kernel avoiding given cyclic subgroup
Let $G$ be a finite non-abelian group and $g \in G$ such that $\langle g \rangle$ is not normal in $G$. I am wondering the following:
Does there always exists a complex irreducible representation $\...
3
votes
1
answer
452
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Lower bound for trilinear character sum in $\mathbb{Z}_2^n$
Consider the group $\mathbb{Z}_2^n$, equipped it with the following dot product: for $a=(a_1,\dots,a_n)\in \mathbb{Z}_2^n$ and $b=(b_1,\dots,b_n)\in \mathbb{Z}_2^n$, define $a\cdot b :=a_1b_1+\dots+...
- 448