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2 votes
0 answers
101 views

On the irreducible submodules of adjoint representations $\text{ad}^{0}$

Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...
stupid boy's user avatar
2 votes
0 answers
137 views

$p$-adic Banach group algebra

Let $G$ be a discrete group. Consider the Banach $\mathbb{Z}_p$-algebra: $$c_0(G, \mathbb{Z}_p) = \{ F : G \to \mathbb{Z}_p \mid \lim_{g \to \infty} |F(g)|_p = 0 \}$$ with the product given by the ...
Luiz Felipe Garcia's user avatar
2 votes
0 answers
144 views

Zero divisors in the extra-special group algebra $\mathbb{R}[2^{1+6}_+]$

Can you characterize the unit-group of the real group-algebra of the extraspecial plus-type 2-group of order 128? (That is $\mathbb{R}[2_+^{1+6}]$ using Conway's notation.) (Please choose any irrep ...
Eric Downes's user avatar
0 votes
0 answers
116 views

Multivariate polynomial representations of the infinite dihedral group

The presentation given in Wikipedia for the infinite dihedral group is $$\langle r,s\mid s^2 =1, srs = r^{-1}\rangle.$$ Let $[R]$ denote the infinite set of reciprocal partition polynomials $R_n(u_1,...
Tom Copeland's user avatar
  • 10.5k
11 votes
1 answer
159 views

Are all indecomposable $\mathbb{Z}_+$-modules over the character ring of a group, character rings of a subgroup?

A $\mathbb Z_+$-algebra is an algebra $A$ over $\mathbb C$ with given basis $\{v_i\}$ such that $$v_iv_j=\sum_k n_{ijk}v_k,\hspace{10mm}n_{ijk}\in\mathbb Z_{\geq0}.$$ An example of such an object is ...
shin chan's user avatar
  • 301
7 votes
1 answer
404 views

Do rational group algebras have an outer automorphism?

In the article "Automorphism groups of simple algebras and group algebras" (1978), Janusz conjectures the following: The group algebra $\mathbb{Q} G$ for a non-trivial finite group has an ...
Mare's user avatar
  • 26.5k
1 vote
0 answers
78 views

tensor dimension/reshaping group

Consider an $N$ dimensional tensor $T$ using the strided view representation used by PyTorch, i.e. we have a storage vector $S$ projected into $N$ dimensions using a size tuple $s$ and a stride tuple $...
mikeyd's user avatar
  • 11
2 votes
0 answers
98 views

Question concerning relationships between different $p$-modular systems and Brauer character values

Let $(K,\mathcal{O},k)$ be a large enough $p$-modular system, where $\mathcal{O}$ is a complete discrete valuation ring of characteristic zero with unique maximal ideal $J(\mathcal{O})$, algebraically ...
Bernhard Boehmler's user avatar
4 votes
1 answer
273 views

Wedderburn decomposition of special linear groups

$\DeclareMathOperator\SL{SL}\newcommand\card[1]{\lvert#1\rvert}$I want to study about Wedderburn decomposition of group algebra $k\SL(n,\mathbb{F}_p)$ where $k$ is either an algebraically closed field ...
Infinity_hunter's user avatar
15 votes
1 answer
639 views

What is the centralizer of a Young subgroup of $S_n$?

In their celebrated paper "A new approach to the representation theory of the symmetric group. II", Okounkov and Vershik prove that $Z(n-1,1)$, the centralizer of $\mathbb{C}[S_{n-1}]$ in $\...
Alvaro Martinez's user avatar
3 votes
0 answers
317 views

How to go from primitive idempotents in $\text{End}_A(M)$ to primitive idempotents in $A$?

Let $K$ be a large enough finite field, let $A$ be a finite-dimensional $K$-algebra. Moreover, let $M$ be a finitely generated $A$-module and let $M = M_1\oplus ... \oplus M_n$ be a decomposition of $...
Bernhard Boehmler's user avatar
7 votes
1 answer
281 views

Question concerning the coefficients of block idempotents

Let $G$ be a finite group. Let $p$ be a prime number such that $p \mid |G|$. Let Irr$(G)$ denote the set of ordinary irreducible characters of $G$. For $\chi\in$ Irr$(G)$ define $e_{\chi} := \frac{\...
Bernhard Boehmler's user avatar
2 votes
0 answers
82 views

Minimizing the spectral radius of certain elements of group rings

Let $G$ be a finite group. Let $I_{G}$ be the ideal on the group ring $\mathbb{C}[G]$ consisting of elements of the form $\alpha\cdot\sum_{g\in G}g$. Let $\lambda_{n}(G)$ be the minimum spectral ...
Joseph Van Name's user avatar
8 votes
1 answer
198 views

Is there always a simple module whose Green correspondent is a simple module under some conditions?

Let $G$ be a finite group and $KG$ its group algebra over some field $K$ with $\mathrm{char}\ K$ dividing the order of $G$. It's well-known that the Green correspondence is compatible with the Brauer ...
Master Gang's user avatar
3 votes
2 answers
449 views

Is there any simple formula for the character of $S_{n}$ represented by the set of $k$-tuples of $\{1,2,...,n\}$?

I'm interested in the representation theory of symmetric groups. I'm now trying to search for the formula for the characters of $\Omega^{k}$, the set of $k$-tuple of elements of $\Omega$ a set of $n$ ...
gualterio's user avatar
  • 1,013
4 votes
1 answer
324 views

What is the $p$-regular partition corresponding to the sign representation of $S_{n}$ over a field of characteristic $p$?

I'm now interested in the modular representation of symmetric groups. It is well-known that for a fixed prime $p$, there is a bijection between the irreducible representations of $S_{n}$ over a field ...
gualterio's user avatar
  • 1,013
2 votes
0 answers
90 views

Regular conjugacy classes and irreducible representations in the infinite, projective case

Let $k$ be an algebraically closed field and $G$ a (not necessarily finite) group. Let $\alpha\colon G\times G\to k^*$ be a multiplier, meaning that $\alpha(s,t)\alpha(st,r)=\alpha(s,tr)\alpha(t,r)$ ...
geometricK's user avatar
  • 1,903
2 votes
0 answers
81 views

The centralizer and normalizer of products of (SU(n) $\times$ SU(p) $\times$ …) in U(m)

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin} $Consider the special unitary group $\SU(n)$ and the unitary group $\U(m)$. Below I specify a specfic way to embed $...
wonderich's user avatar
  • 10.5k
2 votes
0 answers
111 views

The centralizer and normalizer of products of (Spin(n) $\times \dots$) in U(m)

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$ Consider the spin group $\Spin(n)$ and the unitary group $\U(16)$. Below I specify a specfic way to embed $(\Spin(...
wonderich's user avatar
  • 10.5k
3 votes
1 answer
355 views

The normalizer of SU(n) in U(m)?

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$. Below I specify a specfic way to embed $...
wonderich's user avatar
  • 10.5k
1 vote
1 answer
275 views

The normalizer of $\operatorname{Spin}(2N)$ in $\operatorname{U}(2^{N-1})$?

$\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$ I can show that $$ \U(2^{N-1})\supset \Spin(2N) $$ when $2N > 4$ or a positive integer $N > 2$, so $\Spin(2N)$ can be embedded in $\U(2^...
wonderich's user avatar
  • 10.5k
8 votes
0 answers
251 views

When does a semisimple $\mathbb{C}$-algebra come from a group?

Let $\mathcal{A}$ be a semisimple $\mathbb{C}$-algebra. By the Artin-Wedderburn theorem, it is isomorphic to a direct product of matrix algebras: $$ \mathcal{A} = \prod_{i=1}^m M_{n_i}(\mathbb{C})$$ ...
pitariver's user avatar
  • 297
2 votes
1 answer
160 views

MAGMA-question concerning the transformation of a $kG$ -module $M$ into a right ideal of the group algebra

Let $G$ be a finite group and $k$ be a finite field of characteristic $p>0$ such that $p\mid |G|$. Let $M$ be a $kG$-module which has an embedding $M\hookrightarrow kG^{reg}$ into the regular $kG$-...
Bernhard Boehmler's user avatar
6 votes
2 answers
235 views

Is there a CAS that can solve a given system of equations in a finite group algebra $kG$?

Let $k$ be a finite field with char$(k)=p>0$. Let $G$ be a finite group. Consider the group algebra $kG$. I would like to solve a given system of equations in $kG$. Question: Is there a computer ...
Bernhard Boehmler's user avatar
1 vote
0 answers
107 views

Reference request concerning splitting fields for groups that are related to special symmetric groups

Denote the symmetric group of order $n!$ by $S_n$. Let $H:=S_p$ for an odd prime $p$. Every finite field $k$ is a splitting field $(^*)$ for $kH$, in particular $k:=\mathbb{F}_p$. Questions: Is $k:=\...
Stein Chen's user avatar
3 votes
2 answers
279 views

Searching for theorems characterizing when $O_p(G)$ is trivial / non-trivial

Let $G$ be a finite group. Let $p$ be a prime. Let $O_p(G)$ be the $p$-core of $G$. Are there any theorems known saying something like $O_p(G)$ is trivial, if and only if ... and $O_p(G)$ is non-...
LSt's user avatar
  • 237
3 votes
1 answer
95 views

Question concerning Brauer's second main theorem, Brauer correspondent blocks and blocks covered by nilpotent blocks

A version of Brauer's second main theorem is as follows: Let $G$ be a finite group, $x$ be a $p$-element of $G$, $B\in\mathcal{Bl}(G)$, and $\chi\in$ Irr$(B)$. If $d_{\chi\mu}^x\neq 0$ and $\mu$ ...
Bernhard Boehmler's user avatar
15 votes
4 answers
869 views

What is known about ordinary character values at involutions?

Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$ on $V$. Let $x$ be an involution in $G$. I'd like to ask the following Question 1: ...
Bernhard Boehmler's user avatar
6 votes
0 answers
259 views

Diameter of finite rational matrix groups

Suppose $G$ is a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$. For a set $\mathcal{M} \subseteq G$ that generates $G$, define the $\mathcal{M}$-diameter $\mathit{diam}(G, \mathcal{M})$ of $G$ to be ...
Stefan Kiefer's user avatar
38 votes
0 answers
1k views

Groups whose complex irreducible representations are finite dimensional

By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting. It is easy ...
Benjamin Steinberg's user avatar
4 votes
0 answers
218 views

Conjugacy class representatives for the automorphism group of a finite abelian group

Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$. In fact, it's not important that I have exactly one representative from ...
Matt Ollis's user avatar
5 votes
1 answer
271 views

Do all finite-dimensional division algebras appear as Wedderburn factors of rational group rings?

Suppose that $D$ is a division algebra that is finite-dimensional over $\Bbb Q$, does there exist a finite group $G$ such that one of the factors in the Wedderburn decomposition of $\Bbb Q[G]$ is a ...
Lukas Heger's user avatar
2 votes
0 answers
127 views

Multiplicative subgroups of $GL(V)$ which are almost additively closed

Edit: According to comments of YCor and Vincent, I revise the question.I appreciate their comments: Let $G$ be a group. We say that $G$ is a special group if it is isomorphic to a subgroup $H$ ...
Ali Taghavi's user avatar
1 vote
0 answers
155 views

Deduce Schur-Zassenhaus theorem from Wedderburn-Malcev theorem

Is it possible to derive Schur-Zassenhaus theorem from Wedderburn-Malcev theorem? For the existance part I have the following idea: Let $G$ be a finite Group and $N$ an abelian Hall $p$-subgroup. Let $...
Sven Wirsing's user avatar
17 votes
1 answer
420 views

Freeness of tensor product

Let $G$ be a finite group. Is $\mathbb{Z}G\otimes_{Z(\mathbb{Z}G)}\mathbb{Z}G$ free as a $\mathbb{Z}$-module, where $Z$ denotes the centre?
M. Livesey's user avatar
2 votes
3 answers
545 views

Irreducible representations of $\text{SL}(2, \mathbb{F}_q)$ which don't exist in decomposition?

Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = \text{SL}_2(\mathbb{F})$. The group $G$ acts linearly on the $2$-dimensional vector space $\mathbb{F}^2$ and fixes the origin $0$. ...
user98836's user avatar
2 votes
3 answers
318 views

Realization of irreducible $\mathfrak{S}_d$-modules and the representation theory of Lie algebra

Let $n$ be a positive integer. It is well-known that a method to realize irreducible $\mathfrak{S}_d$-modules is to construct the so-called Specht modules $S^{\mu}$ which are submodules in the so-...
Steven's user avatar
  • 159
4 votes
0 answers
167 views

For $P$ $\mathbb{Z}G$-projective, $\mathbb{Q}\otimes P$ is $\mathbb{Q}G$-free

I'm looking for a proof of a theorem of Swan [1, Theorem 3]: If $G$ is a finite group and $P$ a finitely generated projective $\mathbb{Z}G$-module, then $\mathbb{Q}\otimes_\mathbb{Z}P$ is a free $\...
Todd Leason 's user avatar
1 vote
1 answer
271 views

Rank of a locally free $\mathbb Z[G]$-module

This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free. ...
eddie's user avatar
  • 255
9 votes
1 answer
521 views

Which group algebras in analysis are "true group algebras"?

Let $G$ be a group, $A$ a unital associative algebra over ${\mathbb C}$, and let us call a representation of $G$ in $A$ an arbitrary map $\pi:G\to A$ such that $$ \pi(1)=1,\qquad \pi(a\cdot b)=\pi(a)\...
Sergei Akbarov's user avatar
9 votes
1 answer
522 views

Division algebras over extension fields / reducibility of $G$-modules

Reformulation of the question (see below for the original question): Let $K$ be an algebraic number field and $D$ a finite-dimensional $K$-division algebra. Is there a description of the field ...
Oliver Braun's user avatar
0 votes
0 answers
143 views

Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and every $p'$-...
Sven Wirsing's user avatar
3 votes
1 answer
220 views

Intersection of Maximal Left Ideals with Finite Dimensional Quotient

Let $\Gamma$ be a finitely generated group and let $A=\mathbb{C}[\Gamma]$ be the corresponding group algebra over $\mathbb{C}$. Let $X$ be the set of all maximal left ideals of $A$ and let $X_0=\{I \...
Hans's user avatar
  • 3,031
27 votes
5 answers
3k views

Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?

Let $R$ be a finitely generated ring with identity, $M_n(R)$ the set of $n\times n$ matrices. Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? This should be an elementary ...
yeshengkui's user avatar
  • 1,373
4 votes
0 answers
358 views

Minimal prime ideals of a group ring

Let $R$ be a left Noetherian ring (if you prefer you can just think to $R$ as a skew field, I'll be happy with an answer under that hypothesis) and $G$ be a polycyclic-by-finite (or, if you prefer ...
Simone Virili's user avatar
7 votes
3 answers
617 views

Why/when classification of simple objects is "simple" ? E.g. (unknown) classification of simple Lie algebras in char =2,3...

Classification of simple finite-dim Lie algebras for char >=5 has been accomplished not so long time ago, and char p=2,3 is open problem. I wonder what is known/expected for char p=2,3 ? More vague ...
Alexander Chervov's user avatar
4 votes
1 answer
686 views

Character theory of $2$-Frobenius groups.

This is a crosspost of my (slightly longer) question on MSE since I'm not getting any responses there. Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\left(G/...
Alexander Gruber's user avatar
1 vote
1 answer
248 views

Restrictions of Modules and Dimensions

Let K be a finite field and let R,P be groups (with R a subgroup of P). I know that the irreducible KP-modules have dimensions 1,4 and 16 over K. I have a KP-module M, and I know that M has dimension ...
dward1996's user avatar
  • 295
11 votes
0 answers
287 views

What do Multilinear Forms tell us about Representations?

The last few days I have been calculating whether certain group representations are real, complex, or quaternionic. It is well-known that the type of the representation corresponds to what type of ...
ARupinski's user avatar
  • 5,191
3 votes
0 answers
515 views

What happens geometrically when you take associated-graded (or complete, ...) of a group ring at its augmentation ideal?

I am interested in the following functor from Monoids (in $\text{Set}$) to Graded Lie Algebras (over a fixed field of characteristic $0$). (By "graded" I mean only that my Lie algebras have some ...
Theo Johnson-Freyd's user avatar