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.
230
questions
1
vote
0
answers
69
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 $GL(n, \mathbb{H})$ ...
6
votes
1
answer
215
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 ...
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 $$\...
3
votes
0
answers
160
views
Characters of finite fields - Reference request
Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of ...
1
vote
1
answer
259
views
Chevalley restriction theorem
$\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\Sym{Sym}$I'm having a hard time understanding the proof of Chevalley's restriction theorem given by Humphreys in "Introduction to Lie Algebras and ...
8
votes
0
answers
156
views
An identity for characters of the symmetric group
I am looking for a reference for the identity
$$\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 ...
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 ...
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)...
3
votes
0
answers
165
views
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 ...
2
votes
1
answer
208
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 \...
3
votes
0
answers
132
views
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 ...
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$ ...
6
votes
1
answer
276
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
106
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 ...
5
votes
1
answer
262
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 \...
5
votes
0
answers
344
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|}\...
0
votes
0
answers
43
views
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 \...
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)...
19
votes
0
answers
536
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 ...
1
vote
0
answers
100
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 ...
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 ...
2
votes
0
answers
190
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_{...
11
votes
2
answers
513
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)^{...
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 ...
2
votes
0
answers
116
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 ...
2
votes
0
answers
43
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
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 ...
6
votes
0
answers
312
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 ...
4
votes
0
answers
190
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
299
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 $...
12
votes
2
answers
841
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 $C_1$, $C_2$, $C_2^2$, $S_3$, $D_8$, $Q_8$, $C_2^3$, $...
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 ...
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 ...
7
votes
0
answers
136
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 (...
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
$$ f(x) = \...
0
votes
1
answer
291
views
Lower bound of the largest irreducible character degree of alternating group $A_n$
$\newcommand\cd{\mathrm{cd}}$Let $A_m$ and $A_n$ be two alternating groups and $15\le m+2 \le n$. Denote $\cd_m$ and $\cd_n$ as the largest irreducible character degree of $A_m$ and $A_n$, ...
1
vote
1
answer
175
views
Known estimate for gaussian sum $\sum_{x \in \mathbb{F}_q} \psi( a x^m + b x^n)$?
Let $\mathbb{F}_q$ be a finite field, $\psi$ be a non-trivial additive character over $\mathbb{F}_q$, and $a, b \in \mathbb{F}_q$ constants. Is there any known estimate for the gaussian sum
$$\sum_{x \...
5
votes
2
answers
675
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
$$\sum_{q \leq Q}\frac{q}{\phi(q)}\sum_{\...
4
votes
0
answers
97
views
Is there a cohomological interpretation of the bilinear form arising from Clifford theory?
For this question all groups are finite, and representations are over $\mathbb{C}$. The setup is that we have $N$ a normal subgroup of $G$, with abelian quotient $A$, and an irrep $V$ of $N$ that is ...
3
votes
0
answers
98
views
Terminology for the "natural probability measure" on the set of irreducible characters of a finite group
To be specific: if $G$ is a finite group and $\operatorname{Irr}(G)$ the set of its irreducible characters (over the complex field) then we know that
$$
1 = \sum_{\phi \in {\rm Irr}(G)} \frac{(d_\phi)^...
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 $n\gg 0$, ...
3
votes
0
answers
179
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 ...
2
votes
1
answer
285
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 ...
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
$$
\chi^{(n-|\mu|, \mu)}_{1^{a_1} 2^{a_2} \cdots} = q_\mu(a_1, a_2, \ldots) \equiv q_\mu(1^{a_1} 2^{a_2} ...
2
votes
0
answers
193
views
Subquotients of representations and character
Let $G$ be a group, $K$ an algebraically closed field of characteristic zero and $\rho_1,\rho_2:G\to \mathrm{GL}_n(K)$ be two semi-simple representations. What I would like to be able to determine is ...
21
votes
0
answers
458
views
Is there a "direct" proof of the Galois symmetry on centre of group algebra?
Let $G$ be a finite group, and $n$ an integer coprime to $|G|$. Then we have the following map, which is clearly not a morphism of groups in general: $$g\mapsto g^n.$$
This induces a linear ...
2
votes
1
answer
125
views
Existence of universal bound related to characters
Let $G$ be a finite group and $\lambda \in \text{Irr}(G)$ an irreducible complex character of $G$.
Let $m(\lambda) := \min \{ \vert G : H \vert \mid H \leq G, \lambda\vert_H \text{ has a linear ...
11
votes
1
answer
205
views
A question about the adjoint of the Adams operations on representation rings
Let $G$ be a finite group, and $R(G)$ its representation ring over $\mathbb{C}$. We have the Adams operations $\psi^k:R(G)\rightarrow R(G)$, given on the level of characters by: $$\chi_{\psi^k{V}}(g)=\...
3
votes
0
answers
123
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 ...
6
votes
1
answer
269
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 =...