Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

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3
votes
2answers
241 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
0answers
82 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
1answer
143 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
0answers
67 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
1answer
77 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 ...
2
votes
0answers
86 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 ...
5
votes
1answer
102 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
1answer
174 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
0answers
95 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
1answer
112 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
0answers
176 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
0answers
115 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
1answer
172 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
1answer
56 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 ...
10
votes
1answer
165 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
1answer
358 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
1answer
69 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 ...
0
votes
1answer
64 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
1answer
154 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
1answer
110 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
1answer
210 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
0answers
119 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
2answers
141 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
0answers
31 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
0answers
70 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 ...
7
votes
1answer
246 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 ...
3
votes
0answers
82 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
0answers
67 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
1answer
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
0answers
101 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 ...
19
votes
0answers
345 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
1answer
106 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 $$ ...
2
votes
0answers
86 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 ...
2
votes
0answers
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
1answer
103 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
1answer
520 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
0answers
77 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
1answer
121 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 ...
2
votes
0answers
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
3answers
666 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 ...
2
votes
0answers
74 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
1answer
94 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
0answers
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
0answers
115 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
1answer
92 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 ...
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vote
0answers
56 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}, ...
3
votes
1answer
126 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
1answer
139 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
2answers
167 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
1answer
149 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 ...