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

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7
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1answer
218 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 ...
22
votes
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 ...
-1
votes
1answer
126 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 ...
13
votes
1answer
410 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 ...
1
vote
0answers
81 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
votes
1answer
77 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\ ...
2
votes
1answer
144 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 $a,b\in A$, ...
4
votes
0answers
108 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 ...
4
votes
0answers
212 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 ...
15
votes
1answer
278 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 ...
1
vote
0answers
111 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 ...
4
votes
1answer
146 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
votes
1answer
118 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 ...
0
votes
0answers
55 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 ...
2
votes
0answers
117 views

representations of dihedral group/quaternion group of order 8

Is there a classification of such representations via unitriangular matrices over characteristic two fields?
7
votes
1answer
144 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 ...
2
votes
0answers
101 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
votes
1answer
141 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 ...
1
vote
1answer
72 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
votes
2answers
200 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 $\mathbb{C}$ ...
6
votes
1answer
164 views

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 ...
3
votes
0answers
92 views

Annihillator of the highest weight vector in a finite-dimensional representation

Let $\mathfrak g$ be a simple complex Lie algebra and let $V(\lambda)$ be a finite-dimensional representation with highest weight $\lambda$. Let $v$ be the highest weight vector. Then the action of ...
6
votes
2answers
470 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 ...
0
votes
1answer
65 views

Nagakami behavoir

Is the sum of square Nagakami random variables Erlang distributed? What is the distribution of euclidean norm of complex Nagakami? Cheers!
1
vote
1answer
175 views

Does Schur's Lemma hold in this case? Regular representations of $S_n$ over $\mathbb R$

Warning I am a physicist and I am not familiar with a lot of the machinary of representation theory. I consider the regular representation of $\mathbb S_n$ over reals $\mathbb R$ ($\mathbb R \mathbb ...
2
votes
1answer
184 views

Various definitions of the Bruhat decomposition of the affine Grassmannian

Let $G$ be a connected, simply connected, complex, semi-simple group with affine Grassmannian $\mathcal Gr \cong G(\mathbb C[t,t^{-1}])/G(\mathbb C[t])$. Fix a choice of maximal torus $T \subset G$ ...
0
votes
0answers
65 views

Generalized weight space

In their paper Lepowsky and Mcmollum sketch theory of weights in a more general setting. Here is their definition of a weight space: If $A$ is a subset of $\mathfrak g$ and $\lambda$ is a function ...
1
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0answers
51 views

Classification properties of fusion rings

Fusion rings have so many classification properties (I checked the literature a bit) that my head hurts. For practical reasons I define the following three new properties (which might coincide with ...
1
vote
0answers
40 views

Anti-Invariant Polynomials of the Dihedral group

I'm interested in the one-dimensional irreducible representations of $D_{2n}$ acting on $\mathbb{R}[x,y]$. I have found that the trivial representations for an algebra freely generated by $x^2+y^2$ ...
3
votes
0answers
122 views

Invariant Theory over finite adeles

Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$. I am ...
1
vote
1answer
131 views

Which $\frak{sl}_2$-Representations Arise From Hermitian Metrics

Recal that $\frak{sl}_2$ is the Lie algebra with basis elements $e,f,h$, and bracket $$ [e,f] = h, ~~~ [h,e] = 2e, ~~~ [h,f] = -2f. $$ For $M$ a $2n$-complex manifold, the Lefschetz identities tell us ...
2
votes
2answers
153 views

Projection formula for smooth representations of locally profinite groups

Let $G$ be a locally profinite (i.e., locally compact, Hausdorff, and totally disconnected) topological group, $H \le G$ a closed subgroup, $(\pi, V)$ a smooth representation of $G$, and $(\sigma, W)$ ...
1
vote
0answers
79 views

Fundamental invariants for root subsystems

Let $\Phi$ be an irreducible root system of rank $\ell$. The fundamental invariants of $\Phi$ is a set of $\ell$ integers $d_1, \cdots, d_\ell$ canonically attached to $\Phi$. Now suppose $\Psi$ is ...
3
votes
4answers
571 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 ...
4
votes
1answer
159 views

Shift-invariant symmetric functions in representation theory?

The connection between symmetric functions and representation theory is well-known. Now consider the subspace of symmetric functions that are shift-invariant, that is, functions satisfying ...
7
votes
1answer
230 views

On unramified p-adic groups

Let G be a reductive group over a local field F. Let O be the ring of integers of F. The following are equivalent (and groups satisfying these conditions are called unramified): (a) G is quasisplit ...
2
votes
1answer
171 views

Applying Lemma 2.12 of Deligne's/ Milne's “Tannakian Categories” on an irreducible representation

since this is my first question here, I'm not very certain whether I state it properly. I'm thankful for any helpful remarks. Currently I'm trying to understand the above-mentioned article which can ...
2
votes
1answer
82 views

Weight polytopes of the fundamental representations of simple Lie groups

Where can I find a description of the weight polytopes of the fundamental representations of the classical complex simple Lie groups? Thanks in advance
4
votes
0answers
182 views

Decompositions of a compact Lie group into “fixed point set types”

Consider a compact Lie group $G$ which acts on a closed Riemannian manifold $M$ by isometries. Then it is well-known that there are only finitely many isotropy types of the $G$-action, i.e. finitely ...
7
votes
0answers
144 views

What's the appropriate notion of a Unitary representation of a Lie algebra?

Here Lie algebras/groups are real. The most straightforward definition might be: Def: A representation $\rho:\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is unitary if $V$ is equipped with a Hermitian ...
1
vote
1answer
153 views

Homomorphisms from irreducible spaces to reducible spaces

Let $P_{\lambda}$ be a Young symmetriser associated to the following tableau $(a_1 a_2 a_3 b_3 ; b_1 b_2)$ where the entries seperated by the ; belong to first and second COLUMNS of the tableau. Take ...
3
votes
0answers
118 views

Unipotent representations of SL(2,R) by quantization

I'm a PhD student in mathematical physics and I happen to need some elements of Kirillov's "orbit method" for producing representations of Lie groups. I'm familiar with symplectic geometry, geometric ...
7
votes
2answers
126 views

Can monomial representations induced from nonmonomial representations?

Let $H$ be a subgroup of $G$. Let $\rho$ be an irr representation of $G$ induced from an irr representation $\theta$ of $H$. It is well known that $\rho$ is monomial if $\theta$ is monomial. Is it ...
1
vote
0answers
113 views

Non invertibility of certain integral arising from group action

Let a compact topological group $G$ with invariant measure $\mu,$ acts on a simply connected compact topological space $X$ and $\rho$ is a $n$-dimensional unitary representation of $G$. ...
2
votes
1answer
87 views

No basis change in a fusion ring allowed?

Consider the Fibonacci category F: $A\bigotimes{A}=E\bigoplus{A}$. A study by Study (SCNR :-) already 1890 listed all unital associative algebras (with rank<=4). ...
6
votes
3answers
336 views

Generators of invariant polynomials of semisimple Lie algebra

Suppose $\mathfrak{g}$ is a complex semi-simple Lie algebra. By a theorem of Chevalley, we know that $S(\mathfrak{g})^\mathfrak{g}$, i.e. the $\mathfrak{g}$ invariant polynomials, is generated by $l$ ...
2
votes
1answer
95 views

A quadratic algebra with four generators and four relations

Algebra that I'm going to describe pop-up in my research, it looks completely elementary, but I don't know any appropriate references. Let $k$ be an algebraically closed field of characteristic ...
2
votes
0answers
83 views

A question on Plancherel measure for $p$-adic group

Let $G$ be a reductive group over a $p$-adic field. My understanding is that the Plancherel measure on $G$ is a measure on the unitary dual $\hat{G}$. But at the same time, for example, in his famous ...
0
votes
0answers
100 views

G-equivariant coherent sheaves on Bott-Samelson Resolutions

Let $G$ be a Lie group, $B$ be a Borel subgroup. $G/B$ is the corresponding flag variety. Let $w$ be an element of the Weyl group $W$ with a reduced expression $w = s_1 \cdots s_n$. Let $X_w$ be ...
5
votes
2answers
163 views

Weight multiplicity formulae for $(\mathfrak g,B)$-irreps

Let $G$ be a complex reductive Lie group, $B$ a Borel subgroup, with which to define "dominant weight". Let $\lambda$ be an integral weight, not necessarily dominant, but nonetheless giving a ...