**7**

votes

**1**answer

264 views

### What does the unique mean on weakly almost periodic functions look like?

There is a unique invariant mean $m$ on WAP functions on any discrete group (see definitions below, theorem of ?). However, the proofs I found are fairly non-explicit on how to obtain this invariant ...

**4**

votes

**0**answers

93 views

### Short proof of the classification of representation-finite symmetric algebras up to stable equivalence

Assume $K$ is an algebraically closed field and $A$ a finite dimensional $K$-algebra. Assume additionally that $A$ is symmetric and representation-finite.
Then one has the following classification of ...

**4**

votes

**0**answers

68 views

### Efficiently computing all equivariant maps between two $GL_n$-representations

This is sort of a strange question; if it's not appropriate for MathOverflow I apologize in advance.
I'm in a situation where I'd like to be able to give a computer two $GL_n$ representations $V$ and ...

**3**

votes

**1**answer

176 views

### Representations of p-groups where 1 is never an eigenvalue

Fix some $n \geq 1$ and some prime $p$. I'm looking for finite $p$-groups $G$ and finite-dimensional complex representations $V$ of $G$ with the following two properties:
The abelianization of $G$ ...

**4**

votes

**0**answers

111 views

### How do you understand the Moy-Prasad filtration of G_2?

Starting on page 44 of this paper of Reeder and Yu, the authors describe the first graded piece of the Moy-Prasad filtration on $G_2$ at a certain point (in this case it's $GL_2$ of the residue field),...

**19**

votes

**0**answers

350 views

### A cohomology class associated with a complex representation of a group

$\newcommand\CC{\mathbb C}\newcommand\ZZ{\mathbb Z}\newcommand\ad{\mathsf{ad}}\newcommand\Ext{\operatorname{Ext}}$ Suppose that $G$ is a finite group and that it acts on a finite dimensional complex ...

**5**

votes

**1**answer

110 views

### A sum over characters of $S_{2n}$ and zonal spherical functions of $(S_{2n},H_n)$

The hyperoctahedral group $H_n$ can be seen as the centralizer of the permutation $(12)(34)\cdots (2n-1\,2n)$ in $S_{2n}$. It has $2^nn!$ elements.
The quantities $$ \omega_\lambda(\pi)=\frac{1}{2^...

**2**

votes

**0**answers

90 views

### Centralizer of a dense subgroup in a maximal subgroup of a reductive group

I am looking for a reference to the following statement
"Let $G$ be a reductive algebraic group and $K$ a maximal compact subgroup of $G$. If $H$ is a dense subgroup in $K$, then the centralizer of $H$...

**2**

votes

**0**answers

59 views

### Normaliser of image of induced representation

I would like to compute the normaliser of the image of an induced representation. So it would be a representation from $GL_{2}(\mathbb{Z}_{p})$ to $GL_{n}(\bar{\mathbb{Z}}_{p})$. Is this maybe some ...

**4**

votes

**1**answer

109 views

### Connection between representations of different orientations of graph

In 1973 paper about Gabriel's theorem, there is an open question:
Suppose we have a graph $\Gamma$ and two orientations $\Lambda,\Lambda'$ of it. Then for each indecomposable representation of $\...

**26**

votes

**1**answer

538 views

### Why do the adjoint representations of three exceptional groups have the same first eight moments?

For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The ...

**1**

vote

**0**answers

81 views

### What does the regular representation of the coinvariant ring of a unitary reflection group look like?

Let $V$ be a complex vector space of finite dimension $n$ and let $W$ be a finite unitary reflection group. This is, $W$ is a subgroup of $GL(V)$ generated by reflections, i.e., elements $r \in GL(V)$ ...

**1**

vote

**2**answers

154 views

### Antiholomorphic cusp forms of negative weight

Let $k\geq 2$ be an even integer and let $\Gamma=\Gamma_0(N)$. Let $f\in S_k(\Gamma)$. To $f$, one may associate an antiholomorphic cusp form of weight $k$ and level $\Gamma$ by defining $g(z):=f(-\...

**2**

votes

**0**answers

104 views

### Generalization of the sign representation to Hopf algebras

For $G$ a finite group, the sign representation is the one-dimensional representation $\pi : G \to \mathbb{C}$ with $\pi(g)$ the sign of the permutation given by the action of $g$ on $G$ by left ...

**3**

votes

**3**answers

674 views

### Square of non-zero element in group algebra is always non-zero?

Consider a finite group $G$ and complex group algebra $\mathbb{C}(G)$, i.e. formal sums $$ \sum_{g \in G} a_gg, \ a_g \in \mathbb{C},$$ with algebra structure: $$ \sum_g a_gg+\sum_gb_gg=\sum_g(a_g+...

**2**

votes

**0**answers

75 views

### Finite dimensional subrepresentations in a Fréchet space

The Lie group $G := SO(n, 1)$ acts on the hyperbolic space $\mathbb{H}^n$ by isometries. In particular, we have a representation of $G$ on $F := C^\infty(\mathbb{H}^n)$. I assume that $F$ is endowed ...

**0**

votes

**1**answer

95 views

### Whitehead's lemma (Lie algebras) for reductive Lie algebras [closed]

Whitehead's lemma (Lie algebras) is: Let $\mathfrak{g}$ be a semisimple Lie algebra over a field of characteristic zero, $V$ a finite-dimensional module over it and $f$: $\mathfrak{g} \to V$ a linear ...

**4**

votes

**0**answers

171 views

### Compactly supported distributions as a projective G-module

For a Lie group $G$ and a locally convex space $V$ let $\mathcal{E}(G,V)$ be the locally convex space of smooth functions from $G$ to $V$, and accordingly $\mathcal{E}_c^\prime(G,V)$ the space of ...

**2**

votes

**0**answers

118 views

### A representation similar to coadjoint representation?

In a project of quantization, I come up with a finite dimensional representation of $so(d)$ that I wish to find some decent references for it. I guess it could have been studied thoroughly in ...

**3**

votes

**1**answer

93 views

### Are all the Lie bialgebra structure on $sl_n$ coboundary?

In the case of $sl_2$, there are three Lie bialgebra structures. We have three cobrackets $\delta: sl_2 \to \Lambda^2 sl_2$. Each $\delta$ can be written as $\delta=d r$ for some matrix $r$. Therefore ...

**1**

vote

**0**answers

57 views

### Low-dimensional classical r-matrices

Let $g= gl_2$. Suppose that $r \in g \otimes g$ satisfies the following properties:
(1) $r_{12} + r_{21} \in g \otimes g$ is $g$-invariant, $r_{12} = r$, $r_{21} = \tau \ r_{12}$.
(2) $[r_{12}, r_{...

**3**

votes

**1**answer

134 views

### Question absolutely cuspidal representation

Let F be a local field, and $\pi$ an absolutely cuspidal representation of $GL_2(F)$. Then the Kirillov model of the representation is given by the space of locally constant functions with compact ...

**7**

votes

**1**answer

147 views

### Why do we want $p$-permutation modules in splendid equivalences?

First Rickard (in Splendid Equivalences: Derived Categories and Permutation Modules ) and then Rouquier (Block theory via stable and Rickard equivalences, Appendix A.1) define splendid equivalences ...

**2**

votes

**2**answers

171 views

### Recover Poisson bracket on $C[G]$ using the Lie cobracket $\delta: g \to \Lambda^2 g$

By a theorem of Drinfeld, there is a one to one correspondence between Lie bialgebras and Poisson Lie groups. Therefore given a Lie cobracket $\delta: g \to \Lambda^2 g$, there is a Poisson bracket on ...

**2**

votes

**1**answer

152 views

### Classifications of Lie bialgebras

What is the current status of the classifications of Lie bialgebras? In particular, has the following problem been solved? Let $gl_n$ be the general linear Lie algebra. Classify all Lie cobrackets $\...

**3**

votes

**1**answer

142 views

### Cells in affine Weyl groups

This may sound like a very general question, which pretty much reflects my ignorance on the subject.
In the case of Weyl groups $W$, there is a notion of left/right/double cells, which is roughly ...

**2**

votes

**0**answers

146 views

### Deligne-Simpson problem for classical groups

Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation
$$
A_1 +...+A_n =...

**3**

votes

**0**answers

99 views

### Systematic treatment of folding and valued graphs

I'm going to say beforehand that this question has something of a "am I missing something?" flavor. I'm in that odd position mathematicians often find themselves, where a topic has been addressed ...

**2**

votes

**0**answers

89 views

### Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations

I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations:
$[U_i, U_j] = 0$ for $|i-j|>1$
$U_iU_{i+1}U_i=...

**2**

votes

**0**answers

99 views

### Lie Symmetries of the Bessel Differential Equation

The Bessel differential equation has an arbitrary looking form, but a lot is known about it.
$$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $$
Is there a way to derive the Bessel ...

**3**

votes

**0**answers

88 views

### extending local systems on a neighbourhood

Let $Y$ an affine finite type scheme over an algebraically closed field $k$.
Let $S$ be a closed subscheme of $Y$ and $Y'$ the henselization of $Y$ along $S$.
If we have a $\mathbb{Z}_{\ell}$ local ...

**1**

vote

**1**answer

167 views

### l-adic local system. on hensel schemes

Let $k$ be a field, $\ell$ a prime different from the characteristic.
If I take $S$ a closed subscheme of $Y$, which is a $k$-scheme of finite type, is it true that any $\mathbb{Z}_{\ell}$-local ...

**6**

votes

**0**answers

119 views

### Is there a finite test for isomorphisms of rigid monoidal abelian categories, part II

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.)
Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with an additive tensor functor $Rep_G ...

**1**

vote

**1**answer

319 views

### Reference request for an introduction to deformation theory in algebraic geometry

I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...

**12**

votes

**1**answer

190 views

### Is there a finite test for isomorphisms of rigid monoidal abelian categories?

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.)
Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with pair of exact tensor functors $...

**1**

vote

**0**answers

112 views

### Conjugacy scheme, fppf versus GIT

I would be glad to have some guidance in the following.
Let $k$ be an algebraically closed field. Let $G$ be a connected reductive group over $k$. Denote by $\mathfrak{c}$ the Zariski spectrum of the ...

**9**

votes

**1**answer

246 views

### Invariants and stablizers for the $PGL(V)$ action on $End(V\otimes V)$

Let $K$ be a field of characteristic zero, and $V$ a finite dimensional vector space over $K$.
Consider the action of the algebraic group $G:=PGL(V)$ on the vector space $W:=End_K(V^{\otimes 2})$ by ...

**6**

votes

**0**answers

96 views

### Subquotients of Jantzen Filtration for Kac-Moody algebras

Let $\mathfrak{g}$ be a complex symmetrizable Kac-Moody algebra, with triangular decomposition $\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+$. Let $\lambda \in \mathfrak{h}^*$, and $M(\...

**4**

votes

**2**answers

281 views

### odd length Chevalley relations (in rank two)

The unipotent radicals $\text{N}$ of the Borel subgroups of the complex algebraic groups of type $A_2$, $B_2$, and $G_2$ can each be abstractly presented using two one-parameter subgroups $x_1, x_2: \...

**5**

votes

**1**answer

119 views

### Does a $G$-structure on $M$ automatically descend to a contact $G$-structure on $\mathbb{P}T^*M$?

If $M$ is a real smooth manifold of dimension $n+1$, by $D\in\mathbb{P}T^*M$ I mean a tangent hyperplane at some point of $M$. I denote by $b$ the canonical projection of the $(2n+1)$-dimensional ...

**4**

votes

**0**answers

160 views

### Representation theory and associated bundles

I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...

**6**

votes

**0**answers

119 views

### Local character expansion for discrete series representations of $GL_n(F)$

I'm interested about what, if anything, is known about the local character expansion of discrete-series representations of $GL_n(F)$, where $F$ is a $p$-adic field.
First, some notation: let $G$ be a ...

**6**

votes

**1**answer

213 views

### How do I determine a real matrix form for a group representation?

Hello mathoverflow community,
I am a little stucked working on my master thesis. For a representation on $\mathbb{Z}_p\ltimes\mathbb{Z}_p^*$ induced from the additive character $\chi$ of $\mathbb{Z}...

**5**

votes

**1**answer

179 views

### Bott-Samelson construction of a perfect Morse function on G/T

An undergraduate student of mine is interested in writing a "senior" thesis on the topology of Lie groups. Let $G$ be a simply connected compact Lie group and $T$ a maximal torus. One way of ...

**3**

votes

**1**answer

156 views

### Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic

$\newcommand{\al}{\alpha}$
Let $M_n$ be the space of $n \times n$ real matrices.
Question:
For which $n$, is there an inner product on $M_n$ which satisfies:
$$(*) \, \, \langle Q^TXQ,Q^TYQ \...

**7**

votes

**2**answers

236 views

### What kind of geometric object is the Pauli spin matrix vector $\vec{\sigma} = (\sigma_1, \sigma_2, \sigma_3)$?

Physicists routinely wrote all 3 Pauli spin matrices as a vector.
$$ \sigma_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right) \hspace{0.25in}
\sigma_2 = \left( \begin{array}{cc}...

**1**

vote

**0**answers

46 views

### Extra-Lorentzian Kac-Moody algebras

My question is about Kac-Moody (KM) algebras of finite rank with symmetrized Cartan matrices $B = C A$ ($A$ is Cartan matrix) of signatures $(-,-,+,...,+)$, $(-,-,-,+,...,+)$, etc. i.e. with $2$,...

**12**

votes

**1**answer

152 views

### An inequality for the minimal number of generators of a finite group

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating (with $\otimes$ and $\oplus$) the left regular ...

**9**

votes

**3**answers

278 views

### Construct discrete series of SL(2,R) as kernel of twisted Dirac operators

I’m studying the paper of (Baum-Connes-Higson, ex 4.25), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R).
The idea is that each non-trivial ...

**0**

votes

**0**answers

76 views

### the linear span of all matrix coefficients is $C(G,\mathbb{C})$ where $G$ is a finite group

Theorem. Let $\{(R_{\alpha},V_{\alpha})\}$ be a complete set of inequivalent irreducible finite dimensional representations of a finite group $G$. Let $V_{R_{\alpha}}$ be the subspace generated by all ...