**5**

votes

**1**answer

342 views

### On the symmetric group of 2^n elements

Consider the set $ X_1^n=\{1,2,...,2^n\} $. Then define $ X_2^n $ to be the set of two element subsets of $ X_1^n $. I will construct $ X_i $ by induction on $ i $. $ X_i^n $ is the set of two element ...

**1**

vote

**1**answer

139 views

### Orbital integral for matrix coefficients

I am currently aiming at estimating orbital integrals. Maybe surprizingly, I hope for some help in the compact case (ramified places), in proving the usual formula
$$O_\gamma(f) = \int_G f(x^{-1}\...

**6**

votes

**0**answers

94 views

### Modular double of elliptic quantum group

By studying dynamical quantum Yang-Baxter equations and corresponding $RLL$ relations, Felder defined an elliptic version of quantum group $E_{\tau, \eta}(sl_2)$, which can be understood as $\mathfrak{...

**3**

votes

**1**answer

96 views

### Vector fields, diffeomorphism subgroups and lie group actions

Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization:
Let $\{X_j\} \in Vect(M)$ be a ...

**2**

votes

**0**answers

68 views

### Action of orthogonal group on the free Lie algebra

This question is somewhat related and inspired by this post of professor Montgomery.
The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of https://en....

**4**

votes

**1**answer

195 views

### Does there exist finite dimensional irreducible representation of Euclidean or Poincare group in which translation and rotation both act nontrivially?

Does there exist any finite dimensional irreducible rep. of Euclidean or Poincare group in which translation and rotation both act nontrivially?
Let me firstly clarify my question. For example, we ...

**11**

votes

**3**answers

580 views

### Philosophy behind cohomological representations

For a given real reductive Lie group $G$, we have the notion of a representation being cohomological using the Lie algebra cohomology. In particular we know that the discrete series representations of ...

**2**

votes

**0**answers

142 views

### Is there a name for this construction from two representations?

I've come across the following elementary construction in representation theory. Does anyone know if this has a name?
Let $G$ be a group and let $V$ and $W$ be two representations. The tensor ...

**-1**

votes

**1**answer

114 views

### Representations of the $3\times 3$ Heisenberg group [closed]

I am trying to understand how the Heisenberg group is defined because I would like to understand the (irreducible) representations.
Following this article given a symplectic bilinear form $\langle, \...

**5**

votes

**2**answers

148 views

### Cartan subspaces for general algebraic representations

So I feel like asking the following likely open-ended question: What good generalizations of the notion of Cartan subspace do we have?
To be precise, let $G\curvearrowright V$ be an algebraic ...

**7**

votes

**1**answer

191 views

### Hopfian modules

My question is slightly motivated by basic results in linear algebra, for example, that if $F$ is a field then a surjective linear map $F^n \rightarrow F^n$ is injective. More generally, any ...

**4**

votes

**1**answer

230 views

### Is the direct sum in Maschke's Theorem an orthogonal decomposition?

I am reading a paper on coding theory, and it uses a statement, which was claimed to be a reformulation of Maschke's Theorem. But I felt that was false...
Let's say $\mathcal(V):=\mathcal{F}_2^n$ is ...

**11**

votes

**2**answers

340 views

### Can two rational rotations $F_2 = \langle A, B \rangle \to SO(3)$ efficiently approximate the $3 \times 3$ identity matrix?

Let $A,B$ be two rational rotations:
$$ A = \left[\begin{array}{rcc} \frac{3}{5} & \frac{4}{5} & 0 \\
-\frac{4}{5} & \frac{3}{5} & 0 \\
0 & 0 & 1 \end{array}\right]
\quad\...

**9**

votes

**0**answers

215 views

### Geometric Satake and Restriction

The Geometric Satake correspondence (due to Lusztig, Ginzburg, Mirkovic-Vilonen) relates perverse sheaves on the Loop Group $\hat{G}$ (with their convolution product) to the Representations of the ...

**6**

votes

**2**answers

235 views

### What are the special parahoric subgroups in unitary groups?

Let $L$ be a $p$-adic field and let $L'/L$ be a quadratic extension. Let $U_{L'/L}(n)$ be a quasi-split unitary group of $n\times n$ matrices with entries in $L'$. I'm curious about what the special ...

**30**

votes

**4**answers

1k views

### Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?

Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are ...

**7**

votes

**1**answer

178 views

### How fine an invariant of a representation is its quotient singularity?

This is a refinement of a question asked on MSE.
Let $G$ be a finite group and let $V$ be a finite-dimensional faithful complex representation of $G$. Consider $V$ as an affine complex variety. In ...

**4**

votes

**0**answers

106 views

### Does a given bound quiver algebra admit an algebra analogous to the preprojective algebra of a path algebra?

Let $Q=(Q_0,Q_1,s,e)$ be a finite quiver.
In the formal construction of the preprojective algebra associated to $Q$, we set $\ Q^*_1=\{\alpha^*:y\rightarrow x| \alpha\in Q_1, \alpha:x\rightarrow y \}...

**5**

votes

**1**answer

131 views

### Quantum group representations from (convolution) matrix units?

Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$
There is a convolution product on $A=F(\...

**3**

votes

**0**answers

81 views

### How to write down solutions of Yang-Baxter equations for $sl_3$ explicitly?

In the paper, Stolin classifies all quasi-Frobenius subalgebras of $sl_3$. How to write down solutions of Yang-Baxter equations for $sl_3$ explicitly using these quasi-Frobenius subalgebras? Thank you ...

**4**

votes

**1**answer

274 views

### what is the injective hull of indecomposable module of preprojective algebra

Let $Q$ be a ADE type quiver and $s_i$ ($i$ runs through the vertices of $Q$) be the simple $\Lambda$-module with 1-dimensional vector space at vertex $i$ and zero-dim at other vertices. Here $\Lambda$...

**5**

votes

**1**answer

201 views

### Bounding $p$-adic characters and Jacquet-Langlands tranfert

I would like to bound uniformly in $\pi$ the $p$-adic Harisch-Chandra characters $\Theta_p$ for division quaternion algebras. By the Jacquet-Langlands correspondence, it is sufficient to bound it on $...

**16**

votes

**2**answers

359 views

### $G$-action on the integral homology of a compact surface

Let $S$ be a compact connected orientable surface, and let $G$ be a nontrivial finite group acting freely on $S$ and preserving orientation (note the the action being free is a strong condition, since ...

**2**

votes

**0**answers

103 views

### Finite-dimensional representations of DAHA of rank 1

DAHA of rank 1 is defined by the relation
$$
(T - t^{1/2})(T + t^{-1/2})=0~, \quad TXT=X^{-1}~, \quad TY^{-1}T=Y~, \quad
Y^{-1}X^{-1}YXT^2q^{1/2}=1
.$$
To understand its representations, it is useful ...

**2**

votes

**0**answers

148 views

### Could Partial Tiltings be studied as Almost Complete Tiltings?

The first part of what follows is a brief recap of the definitions, setting and motivations for my questions. Experts can find the questions at the end.
Here $k$ denotes an algebraically closed field,...

**3**

votes

**2**answers

262 views

### Borel--Bott--Weil for the Grassmannians

The Borel--Bott--Weil Theorem is usually stated for the complete flag manifold of $SU(N)$. Does an analogue hold for the other flags, for example the Grassmannians?
More precisely, suppose $G(\mathbf ...

**2**

votes

**0**answers

84 views

### Quantum Groups and quantum spaces - From algebra to Analysis

My question will be about the non-standard quantum projective space $\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. I want to see this algebra now on a von Neumann algebraic ...

**6**

votes

**1**answer

152 views

### Super-plethysm?

Let $U$ be a representation of $S_m$ and $V$ a representation of $S_n$. Then the representation $\operatorname{Ind}_{S_m\wr S_n}^{S_{mn}}(U^{\otimes{n}}\otimes V)$ has a nice interpretation in terms ...

**6**

votes

**0**answers

71 views

### 2-functoriality of equivariant derived categories

I am wondering about the 2-functoriality in equivariant derived categories, and I hope that someone can clarify... (apologies if this is a stupid question)
For the more precise formulation, recall ...

**0**

votes

**1**answer

80 views

### presentation for a nilpotent group associated to the square of a coxeter element

This question is related to one asked earlier about inductive presentations of unipotent radicals in Kac-Moody groups.
Let $\Gamma$ be a coxeter diagram --- i.e. an unoriented graph with $r$ vertices ...

**3**

votes

**0**answers

90 views

### Irreducible representations in BGG category $\mathcal{O}$ over (finitely) direct sum of general linear Lie superalgebra

Let $\mathfrak{g} = \oplus_i^k\mathfrak{gl}(m_i|n_i)$ be a direct sum of general linear Lie superalgebras $\mathfrak{gl}(m_i|n_i)$'s with the Cartan subalgebra $\mathfrak{h} = \oplus_i^k \mathfrak{h}...

**5**

votes

**1**answer

109 views

### Explicit formulas for certain elements in $Z(U(\mathfrak{gl_n}))$

Let $\lambda$ be a partition with $\leq n$ rows and let $L_{\lambda}$ be the corresponding irreducible representation of ${\rm GL}_n(\mathbb{C})$. Let $e_m(X_1,\dots,X_n)$ be the $m$th elementary ...

**3**

votes

**1**answer

178 views

### Degree of irreducible representations of a finite cyclic group over $\mathbb{Q}_p$

Let $p$ be a prime and $\mathbb{Q}_p$ denotes the $p$-adic numbers. Is it true that the degree of the nontrivial $\mathbb{Q}_p$-irreducible representations of a cyclic group of order $p^n$ is ...

**0**

votes

**0**answers

98 views

### representations of the special orthogonal group

Consider an $N$-dimensional (algebraic) representation $r$ of the special orthogonal group $SO_m$ over the rational numbers $Q$. Is it true that there exists a representation $\varphi \colon GL_m \to ...

**2**

votes

**1**answer

115 views

### indecomposable modules restricted from $gl_n$ to $sl_n$

Let K be an algebraic closed field, $gl_n$ be the general linear Lie algebra over K, and $sl_n$ be the special linear Lie algebra.
Let $\chi\in gl_n^*$. Let $U_\chi(gl_n)$ be the corresponding reduced ...

**2**

votes

**0**answers

178 views

### Mixed up by definitions of mildly mixing

Here are two setup where the notion of "mildly mixing" comes up: for representations and for group acting by measure preserving transformations (see definitions below). Since a natural class of ...

**2**

votes

**0**answers

118 views

### nilpotent orbits in p-adic period domain

I am going to prove an analogue of the W. Schmid nilpotent orbit theorem in p-adic Hodge theory. It seems to me that a similar limit of filtration by slopes of isocrystals should be semistable. I am ...

**4**

votes

**1**answer

206 views

### inductive construction of unipotent radicals

Consider a directed coxeter diagram $\vec{\Gamma}$, i.e. a finite graph where each edge is decorated with one of the integer weights $\big\{3,4,6\big\}$ and those edges with weights $4$ or $6$ are ...

**3**

votes

**1**answer

57 views

### vector bundles induced by an action of a finite subgroup of $O(n)$

Let $M$ be a path-connected manifold. Let $G$ be a finite subgroup in $O(n)$ and suppose $G$ acts freely on $M$. Then we have an associated vector bundle
$$
\xi(M,G): \mathbb{R}^n\longrightarrow M\...

**10**

votes

**1**answer

172 views

### Meaning of topological tensor products in Frenkel-Gaitsgory

The appendix to http://arxiv.org/abs/math/0508382 by Frenkel & Gaitsgory (following an earlier work of Beilinson) describes three different monoidal structures, denoted by $\otimes^!,\otimes^*,$ ...

**5**

votes

**1**answer

364 views

### analog of Lusztig nilpotent scheme

Fix a quiver $Q$ without loop. Denote the set of vectices of $Q$ by $I$.
Let $\Lambda_V$ be the Lusztig nilpotent scheme with associated vector space $V$ over $I$. Briefly speaking, when $Q$ is a $...

**1**

vote

**1**answer

77 views

### Why are exchange graphs of quivers with the same underlying graph but have different orientations isomorphic?

I know the fact that (undirected) exchange graphs of quivers with the same underlying undirected graph but have different orientations are isomorphic (i.e. quivers that are just finitely many arrow-...

**1**

vote

**1**answer

112 views

### Reference request: $\chi^{\lambda'}(\sigma) = (-1)^{n-\ell(\sigma)} \chi^\lambda(\sigma),$ for characters of the symmetric group

I'm looking for a text I could cite that explicitly states the following result: for $\chi^\lambda$ the irreducible character of the symmetric group indexed by the partition $\lambda$, and for $\sigma ...

**4**

votes

**1**answer

158 views

### Sum of skew characters over hooks and “odd” partitions

Let us call a partition odd if all its parts are odd, and let $Odd(n)$ be the set of all odd partitions of $n$, e.g. $Odd(6)=\{(5\,1),(3\, 3),(3\,1^3),(1^6)\}$.
Let $H(n)$ denote the set of all hook ...

**2**

votes

**1**answer

122 views

### Characterization of restricted weights of representations of real semisimple Lie groups

I need to use the following theorem:
Let $\mathfrak{g}$ be a semisimple real Lie algebra, $\Sigma$ a set of restricted roots for $\mathfrak{g}$. Let $\rho$ be any finite-dimensional representation of ...

**4**

votes

**1**answer

214 views

### The coxeter number condtion in the quantum Lusztig conjecture

This is a question about the second point in Geordie Williamson's answer in
What to do now that Lusztig's and James' conjectures have been shown to be false?
,
which says that the Lusztig ...

**3**

votes

**0**answers

123 views

### reduction mod $p$ of Weyl modules

Let $G$ be a reductive algebraic group defined over a non-Archimedean field $F$.
Let $k_F$ be its residue field, of characteristic $p$.
Assume $G$ is unramified over $F$, then it admits a hyperspecial ...

**6**

votes

**2**answers

148 views

### Can we count the number of simple modules for a reduced enveloping algebra?

Let $G$ be a reductive algebraic group over a field of positive characteristic $p$, which I'll assume to be very good for $G$. Then the Lie algebra $\mathfrak{g}$ is restricted and each simple $\...

**0**

votes

**0**answers

34 views

### The rectification of the transpose of a skew tableau?

Suppose the rectification of a skew tableau is a standard tableau, I want to know if the rectification of the transpose of the skew tableau equals to the transpose of the rectification of the skew ...

**7**

votes

**0**answers

72 views

### Skew zonal polynomials, skew zonal spherical functions, and combinatorics

Zonal polynomials may be expressed in terms of power sums as $$Z_\lambda=n!\sum_\nu \frac{1}{z_\nu }2^{n-\ell(\nu)}\omega_\lambda(\nu)p_\nu,$$ with usual notation in which $\omega_\lambda(\nu)$ are ...