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

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What is the cubic casimir element of sl_3?

I have been thinking about this for some time but have had no luck. I have found some sources that say higher Casimir elements can be obtained by generalizing the second order Casimir, which is ...
2
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0answers
59 views

Newvectors in tensor product representations

Let $\pi_p$ and $\pi_p^\prime$ two smooth admissible irreducible complex representations of ${\rm GL}_2(F)$ where $F$ is a non archimedean local field of residual characteristic $p$ of central ...
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1answer
144 views

Can we say that $A$ is a complement for a group $G$?

Let $A$ be a frobenius complement for a group $G$ i.e. $A$ act on $G$ by automorphism s.t. $C_A(g)=e$ for all nonidentity $g$. Now, Action of $A$ can be linearly extended so that $A$ act on $F[G]$. ...
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1answer
145 views

Powers of traces, integrals over spheres and class functions

I asked this on math.StackExchange a while back but got no answers. I hope I'll be forgiven for the double post. Let $V$ be a complex vector space of dimension $\operatorname{dim}_{\mathbb C} V = n$, ...
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0answers
163 views

Unitary representations of Tarski Monsters and other beasts

Did people study the unitary representations of Tarsky Monsters, for example the ones constructed by Ol'shanskii? Are there any exotic representations, ie. except the ones related to the left regular, ...
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1answer
66 views

Reducible reductive Lie subalgebras of so(p,q)

Is it true that $S(O(p) \times O(q))$ is the only proper subgroup of $SO(p,q)$ of full rank acting on the natural representation $\mathbb{R}^{p+q}$ of $SO(p,q)$ that stabilizes a $p$-dimensional ...
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2answers
320 views

Langlands duality and multiplying cocharacters

Recall that there is a bijection between irreducible representations of a compact real Lie group $G$ and the cocharacters (homomorphisms $U(1) \to G$, modulo conjugation) of the Langlands dual group ...
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1answer
179 views

A question on the representation theory of finite group

By the Burnside theorem, we know that we can decompose the order of a group in to a sum of some integers' square, and these integers are the dimensions of the group's irreducible representations . ...
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1answer
72 views

Reductive subgroup and its derived subgroup with an irreducible represenation

Could you please answer the following question: Let $V$ be a faithful irreducible representation of a connected reductive group $H$ defined over $\mathbb{R}$ Is it true that the derived group of $H$, ...
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107 views

Irreducible representations of $Sp(4,\mathbb{F}_2)$

I'm trying to construct the irreducible representations (over $\mathbb{C}$) of the finite group $Sp(4,\mathbb{F}_2)$. Using GAP, the character table is as follows: $$ \left(\begin{matrix} 1 & 1 ...
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4answers
280 views

Breaking up the free Lie algebra into Gl irreps

The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of https://en.wikipedia.org/wiki/Free_Lie_algebra , the free Lie algebra generated by any choice of ...
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148 views

Rational conjugation of elements of a finite group

Let $G$ be a finite group. Two elements $x$ and $y$ of $G$ are said to be rationally conjugate, written $x \sim_{r} y$, if and only if $\langle x\rangle$ and $\langle y\rangle$ are conjugate subgroups ...
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1answer
145 views

Decomposing representations of finite groups of Lie type via computer

This is related to my previous question here. Let me remind you what that question asked: Let $\text{St}_n(\mathbb{F}_q)$ be the Steinberg module (over $\mathbb{C}$) for ...
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0answers
104 views

Alternate proof of Schur orthogonality relations [migrated]

I am trying to find an alternate proof for Schur orthogonality relations along the following lines. Let $G$ be a finite group, with irreducible representations $V_1$, $V_2$, $\cdots$, $V_d$. Let $V$ ...
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0answers
89 views

Functoriality for non-split orthogonal groups

I am trying to understand the functoriality conjectures of Langlands. We know that the functoriality conjectures imply that automorphic $L$-functions of a connected reductive group are equal to ...
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0answers
174 views

Can one classify irreducible unitary representations of the Weyl algebra?

I saw in this MO post: Is there a machinery describing all the irreducible representations ? that classifying irreducible representations of the Weyl algebra is essentially intractable. My question is ...
5
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1answer
291 views

Representation theory of the general linear group over a finite prime field

I am re-posting a question I asked on math.se here because I am unsatisfied with the answers I obtained. The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely ...
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86 views

Computing $\text{Tor}_*^{R_G} (\mathbb{Z}, \mathbb{Z}) $ for a compact Lie group $G$

Let $R_G$ be the representation ring of $G$ a connected, simply connected Lie group, $I_G$ the augmentation ideal and $\mathbb{Z}=R_G/I_G$. $R_G$ acts on $\mathbb{Z}$ via $V \cdot n = (\dim V ) n$. I ...
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0answers
67 views

About the reduceness of the commuting scheme associated with a symmetric pair

my question is the following one: Let $G$ be a connected reductive algebraic group over the field of complex numbers, and let $V$ be a linear representation of $G$ obtained as the isotropy ...
10
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1answer
200 views

Subquotients in the Verma filtration on Verma modules

Let $\lambda$ be a dominant integral weight of $\mathfrak g$, a finite-dimensional reductive Lie algebra over $\mathbb C$. Let $M(w\cdot \lambda)$ denote the Verma module with high weight $w\cdot ...
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42 views

exponential and anisotropic torus

Let $F$ be a local p-adic field and $G$ a semisimple simply connected group over $F$, $\mathfrak{g}$ its Lie algebra. Let $T$ a maximal anisotropic torus of $G$, split over an etale extension of $F$ ...
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2answers
214 views

Analogue of Borel--Bott--Weil for General Equivariant Vector Bundles

The Borel--Bott--Weil Theorem gives the dimensions of the cohomology groups of the equivariant line bundles over flag manifolds. Does there exist an analogous result for general equivariant vector ...
4
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1answer
147 views

Dirichlet Characters as Eigenvectors

This was asked in Math Stackexchange here but generated no comments or answers. I have slightly edited the original question with the comment in the fourth paragraph and the explicit matrix example at ...
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126 views

Irreducible representations of Weyl group of F$_4$ on zero weight spaces?

This is a follow-up to a recent question here concerning the natural representation of a Weyl group $W$ on the zero weight space of an irreducible representation $L(\lambda)$ of highest weight ...
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109 views

Commutative decomposition for full-rank $A$ and low-rank $B$ matrices that do not commute

1. Motivation Consider symmetric matrices $A,B\in\mathbb{R}^{n\times n}$, and let $A$ be full-rank and $B$ be low-rank. The simultaneous block-diagonalization, defined as the following ...
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3answers
265 views

Restriction of scalars for the adjoint representation of $SL_2(\mathbb F_q)$

Let $p$ be a prime number, $q=p^e$ a power of $p$, and $G=SL_2(\mathbb F_q)$. Let $V$ be the adjoint representation of $G$, i.e. $V$ is the 3-dimensional $\mathbb F_q$-space of of (2,2)-matrices of ...
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0answers
110 views

Unitary dual of $Sp_4(\mathbb{R})$

Do we know the unitary dual of $Sp_4(\mathbb{R})$? If so, can someone provide me any references? How about other rank 2 real groups? Thank you!
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0answers
164 views

On a claim of Deligne about representations of Weil-Deligne groups

In Deligne's article 'Les constantes des equations fonctionelles des fonctions L' http://publications.ias.edu/sites/default/files/Number20.pdf, we find the following claim: Proposition 8.9 (ibid.): ...
6
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0answers
110 views

Resolving ADE singularities by blowing up

Let's say we have a finite subgroup $\Gamma \subseteq SL(2,\mathbb{C})$ and consider the quotient variety $\mathbb{C}^2/\Gamma$, which will have one of the well-known ADE or du Val surface ...
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2answers
2k views

Why can't we take three loops?

Apologies for the vague title and soft question. According to Etingof, Igor Frenkel once suggested that there are three "levels" to Lie theory, which I guess could be given the following names: No ...
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1answer
102 views

In Algebraic Compact Quantum Groups, is an Irreducible Corepresentation equivalent to its Conjugate?

A quantum group $A$ here is an algebraic compact quantum group --- a Hopf*-algebra with a Haar State. Here $\hat{A}$ is the set of linear functionals $\{\mathcal{F(a)}:a\in A\}$ of the form ...
12
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1answer
375 views

Why is Klein's representation of $PSL_2(\mathbb{F}_7)$ hard to obtain?

In his famous article [1] Klein constructs a representation of $G=PSL_2(\mathbb{F}_7)$ in $\mathbb{C}^3$ (of which the first invariant polynomial of three variables gives rise to the famous Klein's ...
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0answers
75 views

Reference for a proof of a projective representation of $A_6$

This question is copied from math.stackexchange, in hope that it might get some attension. I want to understand the proof of There is a projective representation of $A_6 \hookrightarrow ...
3
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1answer
76 views

Idempotents and Structure of Simple GL(n,p) modules in the describing characteristic

Let $F_2$ denote the finite field of two elements, and $GL(n,2)$ the general linear group of degree $n$ over $F_2$. I have a promising line of inquiry into identifying the structure of simple $F_2\ ...
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1answer
86 views

Given a locally nilpotent derivation over a field of characteristic 0 and a local slice, how is the ring homomorphism below defined?

Let $K$ be a field of characteristic 0 and $A$ a $K$-domain. Let $D:A\longrightarrow A$ be a locally nilpotent K-derivation, that is, $D(k)=0$ for all $k\in K$, $D(ab)=(Da)b+a(Db)$ for all $a,b\in A$, ...
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94 views

The representation-theoretic nature of an operator resolvent

Consider parameter $s$ in definition of $R(s,A)=(s I - A)^{-1}$ where $A$ is a linear operator in a vector space $X$. When $X$ is over $\mathbb{C}$, then $s$ is thought to be a complex number. Now ...
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191 views

Faithful and weakly-mixing representations of Property (T) groups in relation to left regular rep

Is it known that: Any countable Property (T) group (or more generally, a non-amenable group) has a faithful, weakly-mixing representation which is NOT weakly included in its left regular ...
13
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1answer
155 views

Weyl group actions on 0-weight spaces

For a complex simple Lie group G with a maximal torus T, we can take a highest-weight representation V of G and look at the 0-weight space, i.e. the subspace of V of elements invariant under T. This ...
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84 views

Clebsch Gordan coefficients of compact quantum groups

Consider a compact quantum group $G$. Let $a, b$ and $c$ be irreducible unitary corepresentations and assume that $c$ is contained in $a \otimes b$. Let $U$ be the intertwiner from the representation ...
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1answer
133 views

Are convolution algebras ever “topologically noetherian”?

For finite groups $G$, we have the group ring $k[G]$, and we can think of $G$-representations as $k[G]$-modules. It is known that for $G$ virtually polycyclic, $k[G]$ is a Noetherian ring, which means ...
9
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1answer
114 views

Representations of $\text{SL}(n,\mathbb{F}_p)$ and $\text{Sp}(2n,\mathbb{F}_p)$ whose dimensions are $p^k$

I should preface this by saying that I am not a representation theorist, so I apologize if this can easily be found in standard sources (but sadly I cannot seem to extract it from any of the books I ...
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0answers
48 views

Integrating new vectors of GL(n,F)

I'd be interested in the following: Let $\pi$ be an irreducible admissible generic representation of $GL(2n,F)$, $F$ a p-adic field. Assume that $\pi$ is ramified and let $W$ be a (non-trivial) new ...
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representations of dihedral group/quaternion group of order 8

Is there a classification of such representations via unitriangular matrices over characteristic two fields?
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1answer
130 views

Lie group actions with only one orbit type, but not defining a principal bundle

Searched-for situation: A compact connected Lie group acts effectively on a closed Riemannian manifold by isometries, such that there is only one orbit type of dimension strictly less than that of the ...
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0answers
97 views

Trivial representation in tensor square [migrated]

Taken from another question in this website. I am not sure why the following statement is true. Suppose $G$ is a group and $V$ an irreducible representation of $G$. One has that ...
2
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0answers
86 views

Generators of the algebra of invariant polynomials on a Lie algebra and the root-space decomposition

Let $G$ be a connected, simply-connected complex semisimple group with Lie algebra $\mathfrak{g}$. Fix a pair $T\subseteq B\subseteq G$ of a maximal torus and Borel subgroup, and let $\mathfrak{t}$ ...
4
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1answer
119 views

Some questions on analytic vectors and the integrability of Lie-algebra representations

I would like to ask a number of questions about the theory of analytic vectors and the integrability of Lie-algebra representations, but before I do so, let me fix the terminology to be used in this ...
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1answer
64 views

Does restriction to an open subgroup preserve projective smooth representations?

Let $G$ be a locally profinite group and $K \le G$ an open subgroup. Does the restriction functor $\mathrm{Res}^G_K$ from the category of smooth $\mathbb{C}$-linear representations of $G$ to smooth ...
2
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2answers
189 views

Compact induction as a tensor product

Let $G$ be a locally profinite (i.e., locally compact Hausdorff and totally disconnected) topological group, $H \le G$ a closed subgroup, and $(W, \sigma)$ a representation of $H$ over $\mathbb{C}$ ...
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Lattice model for Affine Grassmannians of non type A

There is a Lattice model for affine Grassmannians of type A, due to Lusztig. It describes affine Grassmannians of type A as the moduli space of certain subspaces in an infinite-dimensional ...