# Tagged Questions

**3**

votes

**0**answers

100 views

### Parshin's buildings for higher local fields

What is the status of the theory of buildings for higher local fields?
I know that there are some papers of Parshin, in which he describes some examples, like $PGL_2$ and $PGL_3$ over ...

**8**

votes

**0**answers

129 views

### Representations of orthogonal groups vs representations of reflection groups

Let $V$ be a finite dimensional inner product space and $O(V)$ the orthogonal group of $V$.
Let $G$ be a (say, finite) reflection group on $V$, regarded as a subgroup of $O(V)$ ($G< O(V)$.) Let
us ...

**2**

votes

**1**answer

458 views

### 1D TQFT in Freed-Hopkins-Lurie-Teleman

In the first section of Freed-Hopkins-Lurie-Teleman they construct a one-dimensional Topological Quantum Field Theory.
$F(\circ_+)$ is a vector space and $F(\circ_-)$ is the dual.
$F(\circ-\circ)$ ...

**3**

votes

**1**answer

64 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 ...

**9**

votes

**1**answer

152 views

### Fixed set of order p automorphism of Bruhat-Tits tree

I would like to know the structure of the fixed set of an order $p$ automorphism [Edit: induced by a matrix in $GL_2(K)$] on the Bruhat-Tits tree for a p-adic field $K$, specifically in the case where ...

**1**

vote

**2**answers

240 views

### A semisimple group ring

Let $n \in \mathbb{N}$, $p$ a prime number, and $G$ a finite group of order coprime to $p$. Let $R = \mathbb{Z} /p^n \mathbb{Z}$ be the ring of integers mod $p^n$. Must $R[G]$ be semisimple?
As noted ...

**0**

votes

**0**answers

77 views

### Actions and representations of profinite groups

Let $p$ be a prime number, and denote by $\mathbb{Z}_p$ the additive profinite group of p-adic integers. Let $G$ be a finitely generated profinite group of order coprime to $p$, and $V = ...

**3**

votes

**1**answer

148 views

### Integral representations of groups of small order

I have a problem in which it would be helpful to know about the integral representations of some groups of small order (probably of fairly low degree). From what I've gathered so far, cyclic groups of ...

**8**

votes

**2**answers

423 views

### When is the conjugation character almost multiplicity free?

Let me give some motivation, which also explains how I arrived at the question. We may let the finite group $G$ act on itself by conjugation, and this makes the group ring into a $\mathbb{Z}G$-module ...

**2**

votes

**2**answers

180 views

### Minimal *-idempotents for the group algebra of the symmetric group

There is a well-known construction of minimal idempotents in the group algebra of the symmetric group $\mathbb C[S_n]$ using row symmetrizers and column antisymmetrizers. But these idempotents are ...

**1**

vote

**2**answers

351 views

### on the extensions of $ A_5$ by $A_5$ [closed]

Let $G$ be a finite group such that $G$ has a normal subgroup $H$ and $H$ is isomorphic to the alternating group $A_5$. Also we know that $G/H \cong A_5$.
Can we say that $G \cong A_5\times A_5$?
...

**9**

votes

**2**answers

303 views

### Can the difference of non-conjugate pseudoreflections lie in the commutator subgroup?

Let $G$ be a finite group acting on a complex vector space $V$ by pseudoreflections (i.e. every element of $G$ is a product of elements which fix hyperplanes in $V$). I would like to understand the ...

**0**

votes

**2**answers

177 views

### Projective characters with corresponding factor set

The following is just a follow up to my previous question. I have a finite group $H$ with 14 ordinary characters. The Schur multiplier $M(H)\cong 2^2$. Hence the group $H$ will have 3 sets of ...

**4**

votes

**0**answers

136 views

### Is there a notion of “tame” representations of $GL_n(Z)$?

This is a followup to this question about the (left) noetherianity of the group ring of $GL_n(\mathbf{Z})$:
Does GL_n(Z) have a noetherian group ring?
Given that $\mathbf{Z}[GL_n(\mathbf{Z})]$ is ...

**4**

votes

**2**answers

527 views

### Decomposing representations of finite groups

Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}_p$.
Must there be ...

**5**

votes

**2**answers

621 views

### An application of Maschke's theorem

I've been teaching some elementary representation theory to undergraduates, and want to provide applications of Maschke's theorem to complex group algebras to present in class. In particular, I'd like ...

**4**

votes

**3**answers

514 views

### Action of a profinite group

Let $G$ be a finitely generated profinite group, $p$ a prime number. Put $$ V = \prod_{i \in I} \mathbb{Z}_p$$ a (profinite) group equipped with the product topology (for convenience, $I$ may be ...

**4**

votes

**2**answers

114 views

### Reduction of different RG lattices to kG modules

Every book on modular representation theory of finite groups introduces p-modular systems and describes how to reduce an ordinary representation $U$ to obtain one in characteristic p (call it ...

**5**

votes

**1**answer

252 views

### In which fixed-point free representations is the sum of every 3 elements invertible?

A representation $\rho:G\to GL_k(\mathbb{F})$ is called fixed-point free if for every $1\neq g\in G$ and every $0\neq v\in \mathbb{F}^k$, $\rho(g)v\neq v$. Stated differently, it is a representation ...

**1**

vote

**1**answer

192 views

### On the character degrees of a finite group with special structure

Let $G$ be a finite group such that $G$ has a normal subgroup $N$ of order $p(p^2+1)/2$, where $p>13$ is an odd prime and $p\ne 239$. Also $G/N\cong \text{PSL}(2,p)$. Can we say that there exists a ...

**3**

votes

**1**answer

232 views

### A little bit of Intuition for Corepresentations from Representations

I asked this question over on Math.Stack --- where it has a bounty --- but I didn't really get a helpful response so I am asking the question here. One commenter suggests that I am confusing left- ...

**2**

votes

**0**answers

195 views

### The tallest possible lattice?

Let O be a complete discrete valuation ring and G a finite group. Recall that a finitely generated O-free OG-module $M$ such that the traces of the invertible endomorphisms of $M$ generate a strictly ...

**15**

votes

**2**answers

398 views

### Is there a natural notion of completion of a Coxeter system?

Let $(W,S)$ be a Coxeter system. Then any element of $W$ can be written as a finite products of elements of $S$. I want some notion of a "completion" of $W$, call it $\hat{W}$, whose elements are ...

**3**

votes

**1**answer

91 views

### Siegel domains and cuspidal functions

Let $F$ be a number field and $\mathbb{A}$ the ring of adeles over $F$. We consider $P_{n}$ the mirabolic subroup of $GL_{n}$.
Do we have a analog of Siegel subset for the quotient ...

**1**

vote

**1**answer

198 views

### $SO(N^2-1)$ and the adjoint representation of $SU(N)$

It is a known fact that the adjoint representation of $SU(N)$ is a proper subgroup of $SO(N^2-1)$.
I would like to know how a generic $(N^2-1)\times (N^2-1)$ special ($det =1$), orthogonal matrix $O$ ...

**21**

votes

**3**answers

906 views

### How can classifying irreducible representations be a “wild” problem?

Let $q$ be a prime power and $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$-matrices. I've read and heard in several places (see e.g. this mathoverflow question) that classifying ...

**6**

votes

**1**answer

228 views

### Can Galois conjugates of lattices in SL(2,R) be discrete?

Let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$. Suppose that the trace field of $\Gamma$ is a totally real number field of degree $d$. This gives $d$ homomorphisms $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...

**11**

votes

**1**answer

355 views

### Can one explain Tannaka-Krein duality for a finite-group to … a computer ? (How to make input for reconstruction to be finite datum?)

Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the
category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...

**4**

votes

**2**answers

327 views

### Character values bounded away from zero

Character values for a finite group are sums of nth roots of unity. I'm wondering if there are any results bounding nonzero values of irreducible characters away from zero. Or if not are there ...

**3**

votes

**0**answers

139 views

### Finding a basis for the (linear combinations) span of a matrix group, efficiently?

I have an algorithm whose bottleneck is the following task:
Let $\mathbb{F}$ be a finite field.
Given a set of $k$ invertible matrices $g_1,\dots,g_k\in GL_n(\mathbb{F})$, let
$G=\langle ...

**5**

votes

**2**answers

333 views

### Decomposing the conjugacy representation of Sym$(n)$ for small $n$

I am trying to compute the decomposition of the conjugacy representation of some small symmetric groups. Perhaps someone has undertaken a similar calculation.
My own calculations are quite slow, ...

**16**

votes

**1**answer

604 views

### Examples of finite groups with “good” bijection(s) between conjugacy classes and irreducible representations?

For symmetric group conjugacy classes and irreducible representation both are parametrized by Young diagramms, so there is a kind of "good" bijection between the two sets. For general finite groups ...

**2**

votes

**0**answers

69 views

### radical unipotent of a parahoric

Let $G$ a split connected reductive group over $\mathbb{C}$. $F=\mathbb{C}((t))$ and $\mathcal{O}$ the ring of integers.
Let $B$ a Borel subgroup and $I$ the corresponding Iwahori.
Let ...

**4**

votes

**4**answers

434 views

### Structure of the adjoint representation of a (finite) group (Hopf algebra) ?

Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation.
Question 1: what is known about this representation ...

**3**

votes

**3**answers

240 views

### Finding a character of height zero

My character theory is rather weak, so excuse me if this is a triviality.
I have read on the encyclopedia of maths that for any group $G$, every block of $G$ contains an irreducible character of ...

**3**

votes

**3**answers

111 views

### For centralizer subgroups, is the endomorphism ring of a restriction generated by endomorphisms and the centralized element?

In some recent doodlings, I got myself to the point where what I was trying to understand would work out if the following claim were true:
Let $G$ be a group, $g\in G$, and $\rho:G \to ...

**6**

votes

**1**answer

121 views

### Is the kernel of the Bohr compactification minimally almost periodic provided that it is cocompact?

Let $G$ be a locally compact (second countable) group and let
$$
G_0 = \cap \{ \ker\pi : \pi \text{ is a continuous finite-dimensional unitary representation of } G \}.
$$
This is the kernel of the ...

**4**

votes

**1**answer

384 views

### Checking irreducibility

This is related to this question. Suppose I have an $n$-dimensional representation of a finitely generated group, and I want to know whether it is absolutely irreducible. This can, of course, be done ...

**7**

votes

**2**answers

246 views

### Realizing a subgroup of a Lie group as a stabilizer subgroup

Let $G$ a compact semisimple Lie group, $H$ a subgroup of $G$. Is it always possible to find an irreducible representation $R$ of $G$ such that the stabilizer of an $x\in R$ is "locally isomorphic" to ...

**3**

votes

**1**answer

308 views

### Is there any groups $G$ with the property $(*_d)$?

Let $G$ be a finite group of even order has only one non-principal irreducible character $\chi$ of degree $d$, $d\in \mathbb{N}$, with the following property (we name it $(*_d)$):
$(*_d)$: There ...

**2**

votes

**0**answers

119 views

### Do you know any clear classification of groups in which there would exist a unique non-linear character of a given degree?

According to
Lev Kazarin, On Thompson’s Theorem, Journal of Algebra 220, 574–590 (1999)
we know that:
[Corollary 5.3]:Let $$cd(G)=\{\chi(1)|\chi\in Irr(G)\}=\{1,f_1,\dots,f_n,d\}, \;\;n\gt0,$$
...

**4**

votes

**1**answer

154 views

### When does the derived subgroup of $G(F)$ contains the $F$-points of unipotent subgroups of $G$

Let $F$ be a local field of characteristic $0$ and $G$ a connected split reductive group over $F$.
Let's look at the derived groups. We have $(G(F),G(F)) \subset (G,G)(F)$ and this inclusion is of ...

**2**

votes

**0**answers

110 views

### irreducibility of exterior powers

Suppose I have a subgroup $H$ of $GL(V)$ such that $H$ acts irreducibly on all the exterior powers of $V$. Is there any sort of characterization of such things? (I am intentionally not specifying the ...

**11**

votes

**3**answers

704 views

### Does the Alternating group of degree $n>7$ have exactly one irreducible character of degree $n-1$?

We know that the alternating group of degree $n>7$ has an irreducible character of degree $n-1$. The latter number is the smallest nontrivial one for each the alternating group has an irreducible ...

**6**

votes

**3**answers

310 views

### A Hausdorff abelian group with no character?

Pontryagin Duality for locally compact abliean groups gives plenty of continuous (unitary) characters $\chi : A \to \mathbb{R} / \mathbb{Z}$, but if we do not assume local compactness, can anything be ...

**4**

votes

**1**answer

104 views

### Extension of characters of abelian locally compact groups

Let $G$ be an abelian locally compact group and $H$ be its closed subgroup. It is known from Pontryagin duality theory that every unitary character of $H$ can be extended to $G$. I think this is true ...

**8**

votes

**2**answers

250 views

### Are torus knot groups linear?

The fundamental group $T(p,q)$ of the complement of a $(p,q)$-torus knot (in $S^3$) admits the presentation $\langle a, b \mid a^p=b^q \rangle $. Is $T(p,q)$ linear, i.e., is there a faithful ...

**7**

votes

**0**answers

225 views

### Higher-dimensional generalization of Pink's theorem

Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially ...

**2**

votes

**1**answer

176 views

### a question about the semidihedral group?

My question is simple:
If a group $G$ has the same character table with the semidihedral group $SD_{2n}$, are $G$ and $SD_{2n}$ isomorphic ?

**1**

vote

**1**answer

421 views

### Para-Complexification of Lie Groups

Let $G$ be a real Lie group. Then the complexification $G_\mathbb{C}$ of $G$ is the unique complex Lie group equipped with a map $φ:G\to G_\mathbb{C}$ such that any map $G\to H$ where $H$ is a
...