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.

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Characters on ray class groups

Let $K$ be an algebraic number field, $\mathcal{O}_K$ its ring of integers, $\mathfrak{m}$ an integral ideal of $\mathcal{O}_K$. Let $J$ be the set of all fractional ideals, $P$ the set of principal ...
Joshua Stucky's user avatar
1 vote
0 answers
139 views

Efficient decomposition algorithm for characters of symmetric groups

Let $\chi$ be a rational character of $G:=S_n$, and we want to know whether it decomposes into irreducibles $\chi_\lambda$, for $\lambda\in\Lambda$, with $\Lambda$ given, as $$ \chi=\sum_{\lambda\in\...
Dima Pasechnik's user avatar
1 vote
0 answers
82 views

Is there a quaternionic analogue of Weyl's character formula?

I am pondering about this. The definition of a character makes sense for quaternionic matrices. Indeed, given a quaternionic representation of a quaternionic matrix group such as $GL(n, \mathbb{H})$ ...
Malkoun's user avatar
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6 votes
1 answer
222 views

Question about Größencharaktere in imaginary quadratic number fields

Presumably, one could ask this question for a Größencharakter in an arbitrary number field, but I'll restrict my attention to the case I'm interested in. Let $K$ be an imaginary quadratic field with ...
Joshua Stucky's user avatar
3 votes
1 answer
262 views

Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$?

Let $G$ be a finite group generated by permutations $g_1,\dots,g_s$ such that $g_1g_2\cdots g_s$ is the identity permutation. The corresponding Hurwitz representation $V_{\text{Hur}}$ has character $$\...
utx7563yu's user avatar
3 votes
0 answers
160 views

Characters of finite fields - Reference request

Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of ...
Grad Student's user avatar
1 vote
1 answer
261 views

Chevalley restriction theorem

$\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\Sym{Sym}$I'm having a hard time understanding the proof of Chevalley's restriction theorem given by Humphreys in "Introduction to Lie Algebras and ...
Trinity-Slifer 's user avatar
8 votes
0 answers
156 views

An identity for characters of the symmetric group

I am looking for a reference for the identity $$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$ for the irreducible characters of the ...
Hjalmar Rosengren's user avatar
17 votes
1 answer
1k views

Explicit character tables of non-existent finite simple groups

In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...
Sebastien Palcoux's user avatar
5 votes
1 answer
296 views

Conductor of determinant of a 2-dimensional Galois representation divides conductor of representation

I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I'm stuck trying to prove that $N(\det\rho)$ divides $N(\rho)...
ECD56163's user avatar
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Abelian characters and odd perfect numbers?

This question is about applications of abelian characters to odd perfect numbers: Context and Definitions: Let $n$ be a natural number and $D_n$ be the set of divisors. We can make this set to a ring ...
mathoverflowUser's user avatar
2 votes
1 answer
210 views

Structural description of Bohr sets in $\mathbb{Z}_N$

Definition 1. Let $\Gamma\subset \widehat{G}$ and $\delta\in [0,2]$. The Bohr set with frequency set $\Gamma$ and width $\delta$ is the set $\text{Bohr}(\Gamma; \delta)= \big\{x\in G: |\chi(x)-1|\leq \...
RFZ's user avatar
  • 298
3 votes
0 answers
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Representation rings of disconnected reductive groups

Let $G$ be a disconnected complex reductive group, or equivalently a disconnected compact real Lie group, the kind treated by Segal in his (first ever, possibly) paper "The representation-ring of ...
Stefan  Dawydiak's user avatar
1 vote
2 answers
174 views

Prove that the ideal of $\mathbb{C}G$ generated by a family of elements $\lbrace p_i\rbrace_{i=1}^n$ is equal to $\mathbb{C}G$

Given a finite abelian group $G$ consider the group algebra $\mathbb{C}G$ and a set $\mathcal{P}=\lbrace p_i\rbrace_{i=1}^n$ of elements of $\mathbb{C}G$. Define $I$ to be the ideal of $\mathbb{C}G$ ...
Marcos's user avatar
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6 votes
1 answer
277 views

Which finite simple groups are rational-relative-real?

A finite group $G$ is called rational if every element $g \in G$ is conjugate to all of its primitive powers $g^a, a \in (\mathbb{Z}/\operatorname{order}(g))^\times$. Analogously, I'll call $G$ real ...
Theo Johnson-Freyd's user avatar
1 vote
0 answers
107 views

Interplay between additive and multiplicative characters of fields

Let $F$ be a countable field. Given a double Folner sequence $(F_N)_{N\in \mathbb{N}}$ in $F$, an additive character $\chi$ (i.e. $\chi: F \to \mathbb{S}_1$ satisfies $\chi(u+v)=\chi(u)\chi(v)$, for ...
User's user avatar
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5 votes
1 answer
267 views

Product of all conjugacy classes

Related to this post of my coauthor Sebastien, I should also mention that one can also prove the following dual result: For any finite group G, the following identity holds: $$ \left(\prod_{j=0}^m \...
Sebastian Burciu's user avatar
5 votes
0 answers
345 views

Adjoint identity on finite nilpotent groups

Let $G$ be a finite nilpotent group. Consider the following (adjoint) identity from [BuPa, Theorem 9.6]: $$\left(\prod_{\chi \in \mathrm{Irr}(G)} \frac{\chi}{\chi(1)} \right)^2 =\frac{|Z(G)|}{|G|}\...
Sebastien Palcoux's user avatar
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0 answers
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A question about algebraic indicator functions

Let $f \in \mathbb{Z}[x]$ and $m,k \in \mathbb{Z}$. Consider the indicator function $g_f : \mathbb{Z} \to \{1,0\}$ given by \begin{align*} g_f(n) = \begin{cases} 1 &\text{if there exists $r \in \...
matt stokes's user avatar
5 votes
0 answers
216 views

Function on $\mathbb{Z}/p^k \mathbb{Z}$ with small Fourier transform?

For $f:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$, define the Fourier transform $\widehat{f}:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$ in the usual way, viz., $\widehat{f}(\xi) = \sum_x f(x) e(-\xi x/p^k)...
H A Helfgott's user avatar
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19 votes
0 answers
544 views

Large values of characters of the symmetric group

For $g$ an element of a group and $\chi$ an irreducible character, there are two easy bounds for the character value $\chi(g)$: First, the bound $|\chi(g)|\leq \chi(1)$ by the dimension of the ...
Will Sawin's user avatar
  • 135k
1 vote
0 answers
101 views

Demazure character (in type A) from Kostant's partition function?

The Kostka coefficients are the coefficients of Schur expanded in monomial basis, i.e., $s_\lambda = \sum_\mu K_{\lambda,\mu} m_\mu$. They are also the coefficients obtained by taking the complete ...
Per Alexandersson's user avatar
0 votes
1 answer
164 views

Sum of weights of an irreducible representation of $U(N)$

Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries. Firstly, I would like to know ...
Blind Miner's user avatar
2 votes
0 answers
192 views

Characters of alternating groups

I am studying certain properties of characters of alternating groups and I have found precisely three characters not satisfying it up to $A_{15}$. These are: A character of dimension $3.696$ of $A_{...
dm82424's user avatar
  • 350
11 votes
2 answers
514 views

Moments of character degrees - is this result new or folklore?

Context $\DeclareMathOperator\cp{cp}\DeclareMathOperator\AM{AM}\DeclareMathOperator\A{A}$For a finite group $G$ and $k\in\mathbb R$, define $$ m_k(G) = \frac{1}{|G|} \sum_{\pi\in\widehat{G}} (d_\pi)^{...
Yemon Choi's user avatar
  • 25.5k
2 votes
0 answers
130 views

Need for "minimal representation" of a symmetric group

I need to construct a representation of a symmetric group $S_n$, in which a character of the conjugacy class $(n)$ (a class of permutations, which are cycles of a maximal possible length $n$) would be ...
V. Asnin's user avatar
2 votes
0 answers
117 views

How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?

It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
IntegrableSystemsEnthusiast's user avatar
2 votes
0 answers
44 views

Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character

Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...
IntegrableSystemsEnthusiast's user avatar
1 vote
0 answers
173 views

Character extension about $Q_8$

Recently, I am studying the book Navarro - Character Theory and the McKay Conjecture. I am trying to solve the following exercise: (Exercise 5.9) Let $G$ be a finite group and $N\unlhd G$, suppose ...
Shi Chen's user avatar
  • 195
6 votes
0 answers
315 views

(CFSG-free) Finite simple groups whose character degrees square divide its order

Let $G$ be a finite group. It is well-known that for all irreducible complex character $\chi$ then $\deg(\chi)$ divides $\lvert G\rvert$. Motivated by some problems with modular tensor categories, we ...
Sebastien Palcoux's user avatar
4 votes
0 answers
191 views

Schur polynomials are polynomials but also sequences on a lattice?

Monomial symmetric polynomials in $n$ variables $x_1, \ldots x_n$ form a natural basis for the space $\mathcal{S}_n$ of symmetric polynomials in $n$ variables and are defined by additive ...
Arnold Mckenzie's user avatar
2 votes
1 answer
300 views

Evaluations of group characters on cosets of subgroups

Let $G$ be a finite group, $H$ a subgroup of $G$ and $g \in G$. Define $$ [gH] = \sum_{h \in H} gh, $$ viewed an element in the group algebra $\mathbb{C}[G]$. Given an irreducible character $\chi$ of $...
Zach H's user avatar
  • 1,899
12 votes
2 answers
846 views

Finite groups with integral character table

The character table of a finite group will be called integral if all its entries are integers. There are $11$ such groups up to order $16$, namely $C_1$, $C_2$, $C_2^2$, $S_3$, $D_8$, $Q_8$, $C_2^3$, $...
Sebastien Palcoux's user avatar
0 votes
1 answer
113 views

Need the proof of Lemma 7.3.7 in "Finite fields: structure and arithmetics" by D. Jungnickel

I am unable to find a copy of "Finite fields: structure and arithmetics" by D. Jungnickel in a library and I would like to read the proof of Lemma 7.3.7 in that book which states that for an ...
user98273535's user avatar
2 votes
1 answer
114 views

Zeroes of characters of general linear group induced from certain characters of parabolic subgroups

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ind{Ind}$My question is about the types of conjugacy classes of $\GL(n,q)$, the general linear group over the finite field with $q$ elements, on which ...
mathseeker's user avatar
7 votes
0 answers
137 views

When is $\operatorname{Out}(G)$ isomorphic to the symmetries of the character table of $G$?

$\DeclareMathOperator\Out{Out}\DeclareMathOperator\CTS{CTS}$The outer automorphism group $\Out(G)$ of a finite group $G$ act on the character table by permuting columns (conjugacy classes) and rows (...
Davi Costa's user avatar
0 votes
0 answers
68 views

Fourier coefficient of close functions

Let $p$ be some prime. Let $\mathbb{Z}_p$ be the cyclic group of order $p$. Let $f, g \colon \mathbb{Z}_p \to \{\pm 1\}$ be two functions. Recall that the Fourier transform is defined as $$ f(x) = \...
Napoleon's user avatar
0 votes
1 answer
293 views

Lower bound of the largest irreducible character degree of alternating group $A_n$

$\newcommand\cd{\mathrm{cd}}$Let $A_m$ and $A_n$ be two alternating groups and $15\le m+2 \le n$. Denote $\cd_m$ and $\cd_n$ as the largest irreducible character degree of $A_m$ and $A_n$, ...
Sun's user avatar
  • 1
1 vote
1 answer
175 views

Known estimate for gaussian sum $\sum_{x \in \mathbb{F}_q} \psi( a x^m + b x^n)$?

Let $\mathbb{F}_q$ be a finite field, $\psi$ be a non-trivial additive character over $\mathbb{F}_q$, and $a, b \in \mathbb{F}_q$ constants. Is there any known estimate for the gaussian sum $$\sum_{x \...
José's user avatar
  • 209
5 votes
2 answers
676 views

Specific application of Cauchy-Schwarz and Large Sieve

Im reading a paper by Matomaki here, and the following is stated (I'm paraphrasing): "By the Cauchy-Schwarz inequality and the large sieve, we have $$\sum_{q \leq Q}\frac{q}{\phi(q)}\sum_{\...
CBagshaw's user avatar
  • 153
4 votes
0 answers
97 views

Is there a cohomological interpretation of the bilinear form arising from Clifford theory?

For this question all groups are finite, and representations are over $\mathbb{C}$. The setup is that we have $N$ a normal subgroup of $G$, with abelian quotient $A$, and an irrep $V$ of $N$ that is ...
Chris H's user avatar
  • 1,854
3 votes
0 answers
98 views

Terminology for the "natural probability measure" on the set of irreducible characters of a finite group

To be specific: if $G$ is a finite group and $\operatorname{Irr}(G)$ the set of its irreducible characters (over the complex field) then we know that $$ 1 = \sum_{\phi \in {\rm Irr}(G)} \frac{(d_\phi)^...
Yemon Choi's user avatar
  • 25.5k
17 votes
0 answers
343 views

Row of the character table of symmetric group with most negative entries

The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this. Is it true that for $n\gg 0$, ...
Sam Hopkins's user avatar
  • 22.7k
3 votes
0 answers
180 views

commutators and characters

Let $x$ be an element in a finite group and let $\chi$ be an irreducible complex character of $G$. It is well-known that $$|\chi(x)|^2=\frac{\chi(1)}{|G|}\sum_{z\in G}\chi([x,z]).$$ The easiest proof ...
Pablo Spiga's user avatar
2 votes
1 answer
287 views

Pólya–Vinogradov like inequality for a character sum with Euler factors

Let $M$ be a large positive integer, $d$ an odd positive integer and $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{R}$. For a non-principal character $\chi_d = \chi$ with modulus $d$, I ...
Melanka's user avatar
  • 577
2 votes
0 answers
73 views

Confusion Regarding Character Polynomials and Dimensions of Irreducible Representations in the Symmetric Group

I am trying to use the Garsia–Goupil formula. Fundamentally, the character polynomial satisfies $$ \chi^{(n-|\mu|, \mu)}_{1^{a_1} 2^{a_2} \cdots} = q_\mu(a_1, a_2, \ldots) \equiv q_\mu(1^{a_1} 2^{a_2} ...
Sam OT's user avatar
  • 560
2 votes
0 answers
193 views

Subquotients of representations and character

Let $G$ be a group, $K$ an algebraically closed field of characteristic zero and $\rho_1,\rho_2:G\to \mathrm{GL}_n(K)$ be two semi-simple representations. What I would like to be able to determine is ...
curious math guy's user avatar
21 votes
0 answers
458 views

Is there a "direct" proof of the Galois symmetry on centre of group algebra?

Let $G$ be a finite group, and $n$ an integer coprime to $|G|$. Then we have the following map, which is clearly not a morphism of groups in general: $$g\mapsto g^n.$$ This induces a linear ...
Chris H's user avatar
  • 1,854
2 votes
1 answer
126 views

Existence of universal bound related to characters

Let $G$ be a finite group and $\lambda \in \text{Irr}(G)$ an irreducible complex character of $G$. Let $m(\lambda) := \min \{ \vert G : H \vert \mid H \leq G, \lambda\vert_H \text{ has a linear ...
Jan's user avatar
  • 83
11 votes
1 answer
205 views

A question about the adjoint of the Adams operations on representation rings

Let $G$ be a finite group, and $R(G)$ its representation ring over $\mathbb{C}$. We have the Adams operations $\psi^k:R(G)\rightarrow R(G)$, given on the level of characters by: $$\chi_{\psi^k{V}}(g)=\...
Chris H's user avatar
  • 1,854

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