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.

Filter by
Sorted by
Tagged with
1 vote
1 answer
64 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 \...
user avatar
  • 177
5 votes
2 answers
593 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_{\...
user avatar
  • 153
4 votes
0 answers
86 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 ...
user avatar
  • 1,557
3 votes
0 answers
77 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)^...
user avatar
  • 23.9k
0 votes
0 answers
52 views

A matrix of character and trace

Let $q$ be a prime power. Let $g$ be a multiplicative generator of $F_{q^2}$, the finite field with $q^2$ elements. Assume that $l$ is a fairly large prime ($>q^4$) dividing $q^{2(q-1)}-1$. Let $\...
user avatar
  • 421
16 votes
0 answers
302 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 $n\gg 0$, ...
user avatar
  • 18.9k
3 votes
0 answers
111 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 ...
user avatar
2 votes
1 answer
224 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 ...
user avatar
  • 529
2 votes
0 answers
65 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} ...
user avatar
  • 498
2 votes
0 answers
124 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 ...
user avatar
20 votes
0 answers
377 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 ...
user avatar
  • 1,557
2 votes
1 answer
109 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 ...
user avatar
  • 83
11 votes
1 answer
181 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)=\...
user avatar
  • 1,557
3 votes
0 answers
74 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 ...
user avatar
  • 140
6 votes
1 answer
230 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 =...
user avatar
3 votes
2 answers
195 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(...
user avatar
  • 105
4 votes
0 answers
212 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 ...
user avatar
  • 161
11 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 $m \geq 2$ ...
user avatar
  • 22.1k
3 votes
0 answers
157 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 ...
user avatar
  • 944
1 vote
0 answers
119 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 ...
user avatar
1 vote
0 answers
125 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 ...
user avatar
  • 913
3 votes
2 answers
169 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 ...
user avatar
  • 913
0 votes
0 answers
44 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 ...
user avatar
  • 29
15 votes
2 answers
606 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 \...
user avatar
1 vote
1 answer
197 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 ...
user avatar
  • 59
3 votes
0 answers
88 views

Positivity of sequences

Totally positive sequences $\lbrace a_n\rbrace_{n\in\mathbb{Z}}$ are defined as those such that the Töplitz matrix $A_{ij}=a_{i-j}$ is totally positive (all its minors are non-negative). An ...
user avatar
2 votes
0 answers
100 views

Action of diagonal automorphisms on the set of irreducible characters of $D_n(q)$

Let $S$ be $D_n(q)$ where $q$ is a prime power. We know that a diagonal automorphism $\phi_h$ of $S$ is of the form $g\mapsto hgh^{-1}$, where $h\in \hat H$ and $\hat H$ normalizes $S$. Note that $\...
user avatar
  • 213
4 votes
2 answers
195 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 ...
user avatar
16 votes
1 answer
553 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 ...
user avatar
  • 23.9k
3 votes
0 answers
75 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 $2$ ...
user avatar
  • 1,557
3 votes
2 answers
327 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, \...
user avatar
  • 529
7 votes
2 answers
680 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 \...
user avatar
  • 5,231
8 votes
1 answer
187 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]!}...
user avatar
2 votes
1 answer
97 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: ...
user avatar
  • 9,887
1 vote
1 answer
73 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 ...
user avatar
  • 1,093
9 votes
2 answers
453 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 ...
user avatar
8 votes
0 answers
187 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 ...
user avatar
7 votes
0 answers
154 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 = \...
user avatar
4 votes
1 answer
177 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 ...
user avatar
  • 1,235
7 votes
0 answers
419 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 ...
user avatar
4 votes
0 answers
53 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 ...
user avatar
6 votes
0 answers
131 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 ...
user avatar
  • 4,005
4 votes
0 answers
154 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 ...
user avatar
  • 41
3 votes
1 answer
254 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): ...
user avatar
1 vote
2 answers
319 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)...
user avatar
4 votes
0 answers
136 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)| ...
user avatar
2 votes
0 answers
178 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 ...
user avatar
6 votes
3 answers
636 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 ...
user avatar
22 votes
3 answers
1k 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)...
user avatar
  • 22.1k
3 votes
0 answers
312 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) $$\...
user avatar