# Tagged Questions

For questions about characters (homomorphisms from a group into the multiplicative group of a field).

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### Factoring the conductor of a Dirichlet character and factoring generalized Gauss sums

Let $m=m_1m_2\cdots m_r$ with $m_i$ positive integers with $\gcd(m_i,m_j)=1$ for $i\neq j$. Given a Dirichlet character $\chi$ modulo $m$ we can define the characters $\chi_i$ (modulo $m_i$) by ...
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### Simplify $\langle \operatorname{Ind}^G_1 1, \operatorname{Ind}^G_H\phi\rangle_G$

Let $G$ be a finite group and $H$ a subgroup. Let $\phi$ be an irreducible character of $H$ and $\mathbb 1$ the trivial character of the trivial subgroup $1$. Let $\langle,\rangle_G$ be the usual ...
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### Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
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### Character of an $\mathbb{R}G$-module constructed from a $\mathbb{CG}$-module

I have been reading Representations and Characters of Groups by Gordon James and Martin Liebeck. I encountered the following construction of an $\mathbb{R}G$-module from a $\mathbb{C}G$-module. ...
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### Why is the sum of irreducible representations nonzero only when the irreducible representation is trivial?

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In section 3, the authors discuss the probability of measuring the irreducible representation ...
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### Fourier transformation of a group

At the beginning of the section 4 of Fast Quantum Fourier Transforms for a Class of Non-abelian Groups, it is said that, ... calculating a Fourier transform for a group $G$ is the same as decomposing ...
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### Character of subgroup of index 2

Let $\chi$ be an irreducible character of a finite group $G$; if $H\leq G$ is a subgroup of index 2, is $Res_H^G\chi$ irreducible? How do conjugacy classes change from $G$ to $H$?
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### characters in semi-direct product.

The character tables of the irreducible representations of $T_d$ and $C_{3v}$ are linked. In the notation on those pages, $A_1$ and $A_2$ are irreducible representations of degree 1, $E$ is degree 2 ...
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### Question on Fourier-Stieltjes transform (in Rudin, p. $15$)

I have a question on an inequality written on the bottom of this page. Let $G$ be a locally compact group and let $\gamma : G \to S^1 \subset \Bbb C$ be a character of $G$. If $\mu$ is a complex ...
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### Show that over finite fields of characteristic two the finite group $C_3$ has no nontrivial representation

Let $G = \mathbb Z / 3\mathbb Z$ be the cyclic group of order three. I conjecture that every irreducible representation over a finite field of characteristic $2$ must be trivial. But how to prove ...
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### $p$-adic digits via character sums

Let $p$ be a prime and let $n = \sum_{k=0}^\infty n_k p^k$ be a $p$-adic integer with each $0 \leq n_k \leq p-1$. Fix $0 \leq c \leq p-1$. Is there a way to check whether the $i$-th digit $n_i$ equals ...
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### Class function and character of $S_3$ representation

I particularly need help with question 2. The $\textit{character table}$ for $S_3$ is given as follows:\begin{array}{c|c|c|} & \text{1} & \text{(12)} & \text{(123)} \\ \hline ...
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### (Infinite) Non-abelian group with only linear characters

If $G$ is an abelian group, then every irreducible character has dimension one (i.e. is linear), for finite group we also have a converse. Do we have a converse for infinite groups? Or: Does there ...
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### A direct proof that for finite $G$ we have “$G$ abelian iff all irreducible characters are linear”

A finite group is abelian iff all its irreducible characters have dimension one (hence are linear). A common proof uses that the number of irreducible representations equals the number of conjugacy ...