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.
220
questions
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votes
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answer
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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 ...
1
vote
0
answers
71
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 ...
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votes
1
answer
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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:
$$
\left(\prod_{j=0}^m \...
- 837
5
votes
0
answers
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views
Adjoint identity on finite nilpotent groups
Let $G$ be a finite nilpotent group. Consider the following (adjoint) identity from [BuPa, Theorem 9.6]:
$$\left(\prod_{\chi \in \mathrm{Irr}(G)} \frac{\chi}{\chi(1)} \right)^2 =\frac{|Z(G)|}{|G|}\...
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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 \...
- 669
-1
votes
1
answer
159
views
Character tables of semidirect products on Sage
I am trying to find the character table of a semidirect product of two group with Sage. If I try the following I get an error.
...
- 226
5
votes
0
answers
213
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)...
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0
votes
0
answers
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Existence of a specific family of functions on an abelian group with vanishing properties on rank 2 subgroups
Fix a prime $p$, and let $W_0\subset W$ be an inclusion of a codimension one $\mathbb{F}_p$ vector spaces. Let $W_e$ denote a fixed nontrivial coset of $W_0$ in $W$.
The question is whether there ...
- 1,759
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votes
0
answers
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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 ...
- 129k
1
vote
0
answers
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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 ...
- 15.4k
0
votes
1
answer
123
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 ...
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votes
0
answers
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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_{...
- 226
12
votes
2
answers
497
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)^{...
- 25.1k
2
votes
0
answers
120
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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 ...
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2
votes
0
answers
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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 ...
2
votes
0
answers
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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 ...
1
vote
0
answers
165
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 ...
- 181
6
votes
0
answers
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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 ...
- 25.1k
4
votes
0
answers
177
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 ...
2
votes
1
answer
250
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,
$$
viewed an element in the group algebra $\mathbb{C}[G]$.
Given an irreducible character $\chi$ of $...
- 1,849
11
votes
2
answers
781
views
Finite groups with integral character table
The character table of a finite group will be called integral if all its entries are integers. There are $11$ such groups up to order $16$, namely $C_1$, $C_2$, $C_2^2$, $S_3$, $D_8$, $Q_8$, $C_2^3$, $...
- 25.1k
0
votes
1
answer
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views
Need the proof of Lemma 7.3.7 in "Finite fields: structure and arithmetics" by D. Jungnickel
I am unable to find a copy of "Finite fields: structure and arithmetics" by D. Jungnickel in a library and I would like to read the proof of Lemma 7.3.7 in that book which states that for an ...
- 11
2
votes
1
answer
112
views
Zeroes of characters of general linear group induced from certain characters of parabolic subgroups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ind{Ind}$My question is about the types of conjugacy classes of $\GL(n,q)$, the general linear group over the finite field with $q$ elements, on which ...
- 21
6
votes
0
answers
121
views
When is $\operatorname{Out}(G)$ isomorphic to the symmetries of the character table of $G$?
$\DeclareMathOperator\Out{Out}\DeclareMathOperator\CTS{CTS}$The outer automorphism group $\Out(G)$ of a finite group $G$ act on the character table by permuting columns (conjugacy classes) and rows (...
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0
votes
0
answers
68
views
Fourier coefficient of close functions
Let $p$ be some prime. Let $\mathbb{Z}_p$ be the cyclic group of order $p$. Let $f, g \colon \mathbb{Z}_p \to \{\pm 1\}$ be two functions. Recall that the Fourier transform is defined as
$$ f(x) = \...
- 1
0
votes
1
answer
279
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 $A_n$, ...
- 1
1
vote
1
answer
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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
$$\sum_{x \...
- 219
5
votes
2
answers
656
views
Specific application of Cauchy-Schwarz and Large Sieve
Im reading a paper by Matomaki here, and the following is stated (I'm paraphrasing):
"By the Cauchy-Schwarz inequality and the large sieve, we have
$$\sum_{q \leq Q}\frac{q}{\phi(q)}\sum_{\...
- 153
4
votes
0
answers
96
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 ...
- 1,759
3
votes
0
answers
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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
$$
1 = \sum_{\phi \in {\rm Irr}(G)} \frac{(d_\phi)^...
- 25.1k
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votes
0
answers
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Row of the character table of symmetric group with most negative entries
The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.
Is it true that for $n\gg 0$, ...
- 21.2k
3
votes
0
answers
166
views
commutators and characters
Let $x$ be an element in a finite group and let $\chi$ be an irreducible complex character of $G$. It is well-known that
$$|\chi(x)|^2=\frac{\chi(1)}{|G|}\sum_{z\in G}\chi([x,z]).$$ The easiest proof ...
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2
votes
1
answer
253
views
Pólya–Vinogradov like inequality for a character sum with Euler factors
Let $M$ be a large positive integer, $d$ an odd positive integer and $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{R}$. For a non-principal character $\chi_d = \chi$ with modulus $d$, I ...
- 557
2
votes
0
answers
69
views
Confusion Regarding Character Polynomials and Dimensions of Irreducible Representations in the Symmetric Group
I am trying to use the Garsia–Goupil formula.
Fundamentally, the character polynomial satisfies
$$
\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} ...
- 540
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votes
0
answers
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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 ...
- 4,106
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votes
0
answers
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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 ...
- 1,759
2
votes
1
answer
119
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 ...
- 83
11
votes
1
answer
199
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: $$\chi_{\psi^k{V}}(g)=\...
- 1,759
3
votes
0
answers
112
views
The intersection of the kernels of the real valued irreducible characters of a 2-group
For a $2$-group $P$ (that is, $|P|$ is a power of 2) let $K$ be the intersection of the kernels of the real-valued irreducible characters of $P.$ If the center $Z$ of $P$ is elementary abelian, then ...
- 322
6
votes
1
answer
258
views
Pólya–Vinogradov inequality for Eisenstein integers
The Pólya–Vinogradov inequality asserts that a non-principal Dirichlet character $\chi$ with modulus equal to $q$ satisfies
$$\displaystyle \left \lvert \sum_{N < n < N+M} \chi(n) \right \rvert =...
- 24.3k
3
votes
2
answers
206
views
Equality of subsets of abelian groups
Let $G$ be a finite abelian group, $X$ and $Y$ be two non-empty subsets of $G$ of equal size. Suppose that for each irreducible character $\chi$ of $G$ we have $\sum_{x\in X}\chi(x)=\sum_{y\in Y}\chi(...
- 105
4
votes
0
answers
221
views
Orbits of group representation over $\mathbb{F}_2$
Recently I have been considering a problem that has had me venture into some mathematical territory that is new to me, specifically group representations, words, and character theory. I would ...
- 161
12
votes
2
answers
1k
views
The character table of the symmetric group modulo m
Let $S_n$ be the symmetric group and $M_n$ the character table of $S_n$ as a matrix (in some order) for $n \geq 2$.
Question: Is it true that the rank of $M_n$ as a matrix modulo $m$ for $m \geq 2$ ...
- 24.9k
3
votes
0
answers
167
views
Characterizing polynomials which behave like a logarithm modulo $1$
This is a version of a question I asked in Math StackExchange about two weeks ago, which is still unanswered. (UPDATE: the original question for $R=\mathbb{Z}$ has been finally answered, but its ...
- 976
1
vote
0
answers
124
views
A question about Theorem 2.3.1 in Tate's thesis [closed]
I don't understand how to prove a conclusion in the Theorem.
When k is $p$-adic, the subgroups 1+$p^{v}$, $v>0$, of $u$ $(|u|=1)$ form a fundamental system of neighborhoods of $1$ in $u$, We must ...
- 99
1
vote
0
answers
154
views
Restriction of a line bundle on $G/B$ to a fibre which is isomorphic to $\mathbb{P}^1$
Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$, Borel $B \supset T$ and Weyl group $W$. Set $X:=G/B$ and $C_w:=BwB/B \subset X$ for $w \in W$ the ...
- 471
3
votes
2
answers
299
views
Character which defines canonical bundle on flag variety
Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ and Borel $B \supset T$ defining a set of simple roots $\Delta$. Additionally let $\rho$ be the half ...
- 471
0
votes
0
answers
50
views
Counting the number of generating triples of various types in finite simple groups
I am trying to figure out how specific generating triples in finite simple groups are calculated. My understanding is that it uses Frobenius's formula and character theory. I'm not an expert on ...
- 29
17
votes
2
answers
764
views
The finite groups with a zero entry in each column of its character table (except the first one)
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}$Consider the class of finite groups $G$ having a zero entry in each column of its character table (except the first one), i.e. for all $g \...
- 25.1k
1
vote
1
answer
288
views
Characters of p-adic units
Let $p$ be a prime and denote by $\mathbb{Z}_p^{\times}$ the group of $p$-adic units. Suppose that $\chi$ is a character $\chi: \mathbb{Z}_p^{\times} \rightarrow \mathbb{C}^{\times}$. Then it is well ...
- 59