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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4
votes
2answers
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
1answer
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
0answers
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
2answers
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
2answers
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
0answers
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
3answers
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
3answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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 ...