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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4answers
906 views

Apocryphal Maschke theorem?

This may be totally trivial or wrong. I am just posting this because I am sick and tired of trying to understand this myself and I am sure someone out here can just answer it out of his head in 2 ...
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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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3answers
1k views

How are these two ways of thinking about the cross product related?

I was always bothered by the definition of the cross product given in e.g. a calculus course because it's never made clear how one would go about defining the cross product in a coordinate-free ...
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1answer
155 views

Maximal subgroups of indefinite special orthogonal group SO(p,q)

Can someone answer the following question: Is there any classification of maximal proper Zariski-closed real subgroups of $SO(p,q)$ which are not parabolic, and satisfy the following conditions: ...
4
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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 ...
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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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1answer
304 views

Is there a left-adjoint to the restriction of comodules?

If $f\colon C \to D$ is a homomorphism of coalgebras and $\rho\colon V \to V\otimes C$ is a $C$-comodule, then $(1\otimes f)\rho\colon V \to V\otimes D$ is the comodule restricted to $D$. In e.g. ...
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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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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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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 ...
4
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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
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 $H$, ...
2
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0answers
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 ...
4
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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 ...
6
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2answers
411 views

Characterizing the real analytic Eisenstein series

Consider the classical real analytic Eisenstein series $$ E(z,s)=\left(\pi^{-s}\Gamma(s)\frac{1}{2}\right)\sum_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}}, $$ where $z=x+iy$. We think of $E(z,s)$ as a ...
6
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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 ...
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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 ...
10
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1answer
259 views

Condition on a Hopf operad for tensor product in the base categoy to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...
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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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0answers
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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1answer
525 views

Number of standard Young tableaux with fixed corner entry

For a partition $\lambda=(\lambda_1\geq \lambda_2\geq\ldots\geq \lambda_k)$ of $n$, let the set of standard Young tableau of shape $\lambda$ be denoted by $SYT(\lambda)$ with boxes at $(i,j)$ denoted ...
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 ...
7
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1answer
793 views

Is the Duflo polynomial conjecture open?

Let $G/K$ be a symmetric space. Let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan decomposition, with the odd part $\mathfrak{p}$. It is well known that the algebra of invariant ...
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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 ...
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0answers
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 ...
3
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0answers
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 ...
4
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1answer
291 views

Finite Unipotent Groups: References

It will be a great pleasure for me if one can suggest "Survey Articles" on following topics related to the finite unipotent group $U(n,\mathbb{F}_q)$. (Thanks in advance!!!) The number of conjugacy ...
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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
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 ...
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 ...
22
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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 ...
2
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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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1answer
184 views

$q$-Deformed Quillen–Suslin Theorem for the Quantum Vector Spaces?

Define n-quantum vector space to be the algebra $$ {\mathbb C}_q^n := \mathbb{C}< x_i ~ | ~ i =1, \ldots, N>/<x_i x_j = q x_j x_i ~ | ~ i< j>. $$ For $q=1$, we get the usual polynomial ...
14
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4answers
775 views

Moments of the trace of orthogonal matrices

Let $O_n$ be the (real) orthogonal group of $n$ by $n$ matrices. I am interested in the following sequence which showed up in a calculation I was doing $$a_k = \int_{O_n} (\text{Tr } X)^k dX$$ where ...
5
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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.): ...
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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 ...
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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 ...
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
448 views

Special automorphisms of extraspecial groups

Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ consisting of all ...
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 $a,b\in A$, ...
3
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1answer
203 views

A Zero-Multiplicity Problem Related to Foulkes' Conjecture

I'm a combinatorialist that is interested in estimating multiplicities of irreps of $1^{S_{kn}}_{S_k \wr S_n}$ (the action of symmetric group on uniform partitions). I'm aware of the difficulty (or ...
3
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4answers
487 views

Representations of the two dimensional non-abelian Lie algebra

I would like to know a complete description of the indecomposable representations of the two dimensional non-abelian Lie algebra over the complex numbers. The finite dimensional representations would ...