# 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.

168
questions

**4**

votes

**2**answers

173 views

### Schur positivity of a polynomial

Suppose a polynomial of the form
$$\prod_i^d \sum_j^p x_i^{f_j}$$
clearly symmetric, where $f_j\in \mathbb{N}$. There is a way to find the set of $f$ numbers such that this polynomial is Schur ...

**17**

votes

**1**answer

476 views

### The sum (with multiplicity) of the cubes of irreducible character degrees of a finite group

Throughout $G$ is a finite, non-abelian group.
$\DeclareMathOperator\Irr{Irr}\DeclareMathOperator\AD{AD}\DeclareMathOperator\cp{cp}\newcommand\card[1]{\lvert#1\rvert}$
Let $\Irr(G)$ be the set of ...

**3**

votes

**0**answers

64 views

### “Character” theory via dualisable $2$ categories

One interesting way to describe the ordinary (over $\mathbb{C}$) character theory of finite groups is to view the categories $Rep(G)$ together in a $2$ category with bimodules as morphisms. This $2$ ...

**3**

votes

**2**answers

284 views

### Dedekind Zeta functions of Biquadratic fields

Let $F/ \mathbb{Q}$ be a biquadratic field of Galois group $C_2 \times C_2$. Then I know that the Dedekind Zeta function of $F$ can be factored into $L$-functions as; $$\zeta_F(s) = \zeta(s) L(s, \...

**6**

votes

**2**answers

605 views

### Proofs of a character identity?

Let $G$ be a finite group, $g \geq 0, k\geq 1 $ integers, and $(C_1,...,C_k)$ a tuple of conjugacy classes of $G$. I am interested in proofs of the following identity
$$
\sum_{(c_1,...,c_k) \in C_1 \...

**8**

votes

**1**answer

155 views

### sl(2)-reps categorifying q-binomials

Recall that the $q$-binomial coefficient $\big[\begin{smallmatrix}a\\b\end{smallmatrix}\big]$ is the Laurent polynomial in $q$ given by
$$
\big[\begin{smallmatrix}a\\b\end{smallmatrix}\big]=\frac{[a]!}...

**2**

votes

**1**answer

88 views

### Source for: a permutation group is multiplicity-free if and only if its 2-orbits define an association scheme

I have recently proven the following (at least, so I believe):
Theorem. Given a permutation group $\Sigma\subseteq\mathrm{Sym}(\Omega)$ on the set $\Omega:=\{1,...,n\}$, the following are equivalent:
...

**1**

vote

**1**answer

64 views

### Are top Brauer characters bounded?

Let $p_\lambda$ be power sum symmetric functions. Let $s_\lambda$ and $o_\lambda$ be irreducible characters of the unitary and orthogonal groups $U(N)$ and $O(N)$, respectively (the $s$ are the Schur ...

**9**

votes

**2**answers

435 views

### Characterization of the family of simple groups PSL(2,q) by tensor multiplicity

Let $G$ be a finite group and $(\chi_i)$ its irreducible characters. Then $\forall i,j,k, \exists!n_{i,j}^k \in \mathbb{N}_{\ge 0}$ such that $$\chi_i \chi_j = \sum_k n_{i,j}^k \chi_k.$$
Let the ...

**8**

votes

**0**answers

167 views

### Question on calculating character sums

I am wondering if there are any references that would help me with the following problem:
Let $p>2$ be a prime number, $n \in \mathbb{Z}^+$ odd such that $(n,p)=1$, $\chi$ an imaginary quadratic ...

**7**

votes

**0**answers

114 views

### Limit of the Casselman–Shalika Formula for the Spherical Whittaker Function

$\DeclareMathOperator{\GL}{GL}$Consider $G = \GL_{r+1}(F)$, where $F$ is a local non-archimedian field with the ring of integers $\mathcal{O}_F$ and the maximal ideal $\mathfrak{p}$, and let $q = \...

**4**

votes

**1**answer

166 views

### Generalization of $\lim_{n \rightarrow \infty} \prod_{i=1}^{n}\frac{2i-1}{2i}$ for a character $\chi:\mathbb{Z}/s \mathbb{Z} \rightarrow \mathbb{C}^*$

Playing with some infinite products I came up with this problem, that I'm not able to figure it out by myself. Moreover in the internet it doesn't seem to appear anywhere.
Maybe it is just an easy ...

**7**

votes

**0**answers

406 views

### Is there a name for these kinds of polynomials?

I've come across the following polynomials in my research and I am wondering if they have a name or if there is very much known about them:
\begin{equation}
F_{\chi}(T) = \sum_{a = 1}^{n-1} \chi(a)T^a
...

**4**

votes

**0**answers

45 views

### Color algebras and color involutions

If $A$ is a $G$-graded algebra then one can define on it a color involution, i.e. a bijective linear map preserving the grading such that the image of a product of two homogeneous elements is defined ...

**5**

votes

**0**answers

84 views

### Representations of 2-groups and quantum double constructions

Let $G$ be a finite group. The category of its representations (complex linear, finite dimensional, throughout this whole question) is equivalent to $\mathbb{C}[G]$-modules. V. Drinfeld constructed a ...

**4**

votes

**0**answers

116 views

### Indexed character tables for wreath products in Sage and GAP

I am trying to obtain character table for the Hyperoctahedral group $\mathcal{H}_n$ in Sage using GAP.
This group arises as the wreath product $\mathcal{C}_2 \wr \mathcal{S}_n$, so of course I can ...

**3**

votes

**1**answer

195 views

### A global code for the character table of PSL(2,q)

We can easily get the character table of $\mathrm{PSL}(2,q)$ for some fixed small prime power $q$, we can just do (for example):
...

**1**

vote

**2**answers

295 views

### Are the character degrees determined by the conjugacy class sizes?

The computation below (part 1) shows that if two finite groups of order at most $100$ have the same (ordered) list of conjugacy class sizes, then they also have the same (ordered) list of (irreducible)...

**4**

votes

**0**answers

133 views

### Characters, centralizers and cosets

I am trying to understand/count the number of solutions of a number of equations in finite groups and came across the following class function:
$$ \theta_\chi(x) = \sum_{y\in G} |C_G(xy)y \cap C_G(x)| ...

**2**

votes

**0**answers

85 views

### Link between characters and isotypic components

I am currently studying finite complex reflection groups using the book written by Lehrer and Taylor called "Unitary Reflection Groups" and I am unsure if I understood isotypic components ...

**6**

votes

**3**answers

438 views

### The zero entries in the character table of a finite group

When you browse the character tables of the small finite groups (for example here), you can observe that every zero entry corresponds to the value of an irreducible character $\chi$ on a non-central ...

**21**

votes

**3**answers

996 views

### On permanents and determinants of finite groups

$\DeclareMathOperator\perm{perm}$Let $G$ be a finite group. Define the determinant $\det(G)$ of $G$ as the determinant of the character table of $G$ over $\mathbb{C}$ and define the permanent $\perm(G)...

**3**

votes

**0**answers

284 views

### Character table of the symmetric group $S_n$ according to James

In James, "The representation theory of the symmetric groups" an algorithm is described to produce the character table of a symmetric group. The proof involves the equation (pp. 22,23)
$$\...

**3**

votes

**0**answers

267 views

### Closed form for Jacobi sum $\sum_{a\in \mathbb{F}_{p^2}}\chi(a)\chi(1-a)$

Let $\mathbb{F}_{p^2}$ be a field with $p^2$ elements and $\chi:\mathbb{F}_{p^2}^*\to\mathbb{C}^{*}$
be a multiplicative and non-trivial character on the multiplicative group $\mathbb{F}_{p^2}^*$ (...

**4**

votes

**1**answer

325 views

### “Simple” proof of irreducible characters of finite groups being non-zero

A search brought up this, with reference to a book by I. M. Isaacs. However, the proof in the book leverages on a lot of field theory knowledge. I am wondering, is there a simpler proof (or a proof ...

**1**

vote

**0**answers

42 views

### self dual character of local fields and global fields

There are two concepts of self dual character, one is for global and another is for local.
Let $K$ be an imaginary quadratic number field, and a Hecke character $\chi : \mathbb{A}_K^{\times}/K^{\...

**1**

vote

**1**answer

80 views

### Restriction of real irreducible 2-Brauer characters to subnormal subgroups

Question: Find a finite group $G$, a subnormal subgroup $H$ of $G$, a real-valued irreducible $2$-Brauer character $\chi$ of $G$ and a real-valued irreducible $2$-Brauer character $\mu$ of $H$ such ...

**1**

vote

**0**answers

61 views

### Relationship between non-zero values of characters and normality in finite groups

Note: Let $G$ be a finite group and $H \leq G$. Then it is clear that if $H \unlhd G$, then $\chi(h) \neq 0$ for irreducible constituents $\chi$ of the permutation character $(1_H)^G$ and $h\in H$. ...

**2**

votes

**0**answers

80 views

### Characters of sets of representations closed under tensor product

Let $R$ denote the set of all irreducible representations of a group $G$ over a complex vector space. Let $U \subset R$ denote a subset of representations which is closed under tensor product (i.e., ...

**3**

votes

**1**answer

305 views

### Representations are determined by characters : Groups and Lie algebras

I know that any finite-dimensional complex representation of a finite group $G$ is determined by its characters. This is immediate, in view of the complete reducibility of this category modules.
My ...

**3**

votes

**1**answer

170 views

### Character values of principal series representations of $GL_n(\mathbb{F}_q)$

Let $P_{\alpha}$ be the principal series representation of $GL_n(\mathbb{F}_q)$, where $\alpha = ( \alpha_1, \alpha_2, \cdots, \alpha_n)$ and $\alpha_i : \mathbb{F}_q^* \rightarrow \mathbb{C}^*$.
...

**2**

votes

**0**answers

161 views

### View Dirichlet character as a character of Galois group

In Jaclyn Lang's article "On the image of the Galois representation associated to non-CM Hida family" section 2, the Dirichlet character $\chi$ module $N$ is also viewed as a character $\chi\colon\...

**9**

votes

**2**answers

457 views

### Character theory and Quantum Chemistry

Who (presumably a chemist) realized first the efficiency of character theory in calculations of orbitals of atoms? In which year?

**0**

votes

**0**answers

108 views

### Definition of “tame $p$-part of $\chi$”

At the beginning of Haruzo Hida's article "Big Galois representations and $p$-adic L functions", he has defined $$\chi_1= \textrm{the } N_0 \textrm{-part of} \chi \times \textrm{the tame } p\textrm{-...

**2**

votes

**0**answers

228 views

### Limit of $\sum (-1)^ {\log \log n }$

Are there results, or can someone point me to reading, on computations of the following sort:
$\lim_n \sum_{2 \le x\le n} (-1)^ { \log \log x }$
It would seem that there is reason to believe this ...

**3**

votes

**0**answers

157 views

### Is the character of the adjoint representation of $\operatorname{SU}(n)$ non-vanishing on regular points of a maximal torus?

Are the maximal and minimal values of the character of the adjoint representation of $\operatorname{SU}(n)$ restricted to a maximal torus known? Can such a character vanish at some regular point of a ...

**2**

votes

**0**answers

52 views

### Super characters in the theory of Lie superalgebras

Weyl character formula for the finite dimensional complex semisimple Lie algebras plays a crucial role in the theory of highest weight modules, where for the highest weight module $V(\lambda)$ we ...

**10**

votes

**0**answers

369 views

### Probability of Words Summing to $1$ in $S_n$ or $PGL_2(n)$

Let $G$ be the symmetric group $S_n$ or the projective general linear group $PGL_2(n)$.
Let $X$ be a cyclically reduced word in the abstract variables
$x_1, x_2, \ldots,x_k$, i.e. $X$ is a product ...

**3**

votes

**0**answers

64 views

### Expressing $\sum_{g\in [G/H]}ge_Hg^{-1}\in Z(\mathbb{C}[G])$ in terms of primitive central idempotents?

Suppose $G$ is a finite group, and $H$ a subgroup. For an irreducible character $\chi$ of $G$, there is a central idempotent in the group algebra $\mathbb{C}[G]$:
$$
e_\chi=\frac{\chi(1)}{|G|}\sum_{g\...

**3**

votes

**1**answer

235 views

### Growth rate of $|{\rm cd}(S_n)|$

The question
What is the order of magnitude for the function $n\mapsto |{\rm cd}(S_n)|$?
The motivation
In my research on character degrees of finite groups, I have in recent years been focusing on ...

**15**

votes

**1**answer

693 views

### Explicit version of the Burgess theorem

Does there exist a totally explicit version of the Burgess theorem? Precisely, let $m$ be a positive integer, and let $\chi$ be a primitive character mod $m$. A special case (sufficient for my ...

**6**

votes

**0**answers

186 views

### Orthogonality relations for characters of VOAs?

If $G$ is a finite group, the characters of its irreps satisfy
$$
\langle \chi_1,\chi_2\rangle := \frac{1}{|G|}\sum_{g\in G} \chi_1(g)\; \overline{\chi_2(g)} = \delta_{\chi_1,\chi_2}.
$$
Alexei ...

**6**

votes

**1**answer

256 views

### Is a finite group given by its character table if its Sylow subgroups are so?

As pointed out by Mikko Korhonen in this answer, Özdem Çelik proved (in 1976 here) that a finite group whose Sylow subgroups are cyclic (called a Z-group) is determined by its character table.
...

**1**

vote

**0**answers

77 views

### Two versions of the Murnaghan-Nakayama rule

I have always see the following Murnaghan-Nakayama rule for a partition $\lambda$ and a permutation $\sigma \in \mathfrak{S}_n$ of cycle structure $(\sigma_1, ..., \sigma_n)$:
$$
\chi_{\lambda}(\sigma)...

**4**

votes

**1**answer

328 views

### Finite groups with the same character table *including* class types, and square-free order

There are non-isomorphic finite groups with the same (complex) character table, as $D_4$ and $Q_8$.
$$\scriptsize\begin{array}{c|c}
\text{class}&1&2A&2B&2C&4 \newline
\text{...

**2**

votes

**0**answers

77 views

### A character sum $\sum_{0<n\leq Y}\chi_4(n)\chi_0^{(k)}(n)$ estimate

I'm reading the paper 'Jutila, Matti. "On the Mean Value of $L (1/2, χ)$ FW Real Characters." Analysis 1.2 (1981): 149-161.'
Let $\chi_4(n)$ be the real primitive nonprincipal character of modulo 4, ...

**1**

vote

**0**answers

29 views

### Formal character and unit

Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$.
...

**5**

votes

**0**answers

105 views

### Does the Gauss sum attached to $\chi$ ever belong to $\mathbb{Q}(\chi)$?

Let $p$ be a prime number and $\chi$ be a primitive Dirichlet character of conductor $p$. We let $$g(\chi)=\sum_{a=1}^{p}{\chi(a)e^{2i\pi a/p}}$$ be the Gauss sum attached to $\chi$. Is this known ...

**2**

votes

**0**answers

134 views

### Does the pushforward of the Haar measure of a semisimple compact Lie group along a character determine the character?

Let $G$ be a connected compact semisimple Lie group.
Let $V$ be a faithful representation of $G$,
with character $\chi \colon G \to \mathbb{C}$.
Let $\mu_G$ be the normalized left Haar measure. (So $\...

**2**

votes

**1**answer

81 views

### Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight

Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight.
It is an exercise of Bröcker's book on Representations of Compact Lie ...