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

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7
votes
1answer
187 views

What is the level of a positive energy loop group representation?

I am trying to learn a bit about loop group representation theory to understand its role in string geometry. Let $G$ be a Lie group. I am thinking of $\text{Spin}(n)$, so you may assume $G$ to be ...
3
votes
0answers
103 views

Symmetric power of an algebra

Given an algebra $A$ over $k$ with characteristic zero and a positive integer $n$, the subspace of $A^{\otimes n}$ consisting of all tensors invariant under the action of all permutations ...
1
vote
1answer
147 views

Name for class of matrix determinants

Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions $\alpha_i(x)$. Now, ...
1
vote
2answers
108 views

When is the induced representation factored through the initial one?

Let $H$ be an open subgroup in a locally compact group $G$, $\iota:H\to G$ the embedding of $H$ into $G$, $\pi:H\to B(X)$ a unitary representation of $H$ in a Hilbert space $X$, and $\rho:G\to B(Y)$ ...
2
votes
0answers
95 views

quasi-split algebraic group [migrated]

While reading papers, there usually an assumption "quasi-split" for reductive algebraic groups. To use their results I need to know which groups are quasi-split. For the case I am interested in ...
5
votes
0answers
359 views

A function canonically associated to an irreducible representation in L^2(M) for a Riemannian G-manifold M. Who has seen it?

The following is my first question here on mathoverflow. Let $M$ be a closed connected Riemannian manifold with an isometric effective action of a compact connected Lie group $G$. Consider the ...
3
votes
0answers
185 views

Reference request: Beilinson-Bernstein for finite-dimensional reps and category O

I think I’ve once been told that under the Beilinson-Bernstein correspondence, finite-dimensional representations of a semisimple Lie algebra $\mathfrak{g}$ correspond to (twisted) D-modules on $G/B$ ...
-1
votes
0answers
62 views

Confusion about the projected component in an irreducible space in the tensor product decomposition using Littlewood-Richardson?

The regular representation of the symmetric group can be formulated in terms of an abstract tensor, where the action of the symmetric group elements is by means of permuting the indices. Given an ...
1
vote
0answers
97 views

Ext Quivers and their applications to Representation Theory

I am looking for references that provide an overview of the following two topics (it can be multiple references if necessary): How to compute the Ext-quiver of a (locally finite or finite) ...
2
votes
2answers
159 views

Jordan-Holder vs Harder-Narasimhan

Let $M$ be a module over an algebra or a group. I am interested the following decreasing filtration: $F^0M=M$; $F^iM$ is the smallest sub-module of $F^{i-1}M$ such that the quotient is ...
2
votes
1answer
148 views

When are induction and coinduction of representations of Lie groups isomorphic? When they are compact? Semisimple?

This is in a sense a follow up on the popular question Induction and Coinduction of Representations, where this particular question is one of several points, and it is neglected. It seems that the ...
2
votes
0answers
45 views

Are Zariski-dense representations of a cocompact complex hyperbolic lattice non-obstructed?

Question Suppose that $\Gamma < \text{SU}(n,1)$ is a cocompact lattice, and let $\rho \colon \Gamma \to G$ be a representation to a non-compact simple Lie group (most interesting case for me: $G = ...
5
votes
1answer
223 views

Proving that the Jones polynomial is q-holonomic

The Jones polynomial is known to have many different interpretations or definitions, by now. There are connections with QFT, quantum groups, Hilbert schemes, Cherednik algebras, etc. My question is ...
5
votes
0answers
181 views

Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
9
votes
3answers
280 views

Maximal compact subgroup of p-adic lie groups

Let $k$ be a number field and $S$ be a finite set of places of $k$. Let $G$ be a connected semisimple algebraic group over $k$. Let $k_S=\prod_{v\in S}k_v$ where $k_v$ is the completion of $k$ at $v$. ...
6
votes
1answer
230 views

An inequality on representations and subgroups of general linear groups over finite field

Let $q$ be a power of $p$, let $l$ be a prime different from $p$, and let $H_1$ and $H_2$ be two subgroups of $GL_n(\mathbb F_q)$ that are $l$-groups. If for all characteristic $0$ representations ...
13
votes
3answers
629 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 ...
3
votes
0answers
56 views

How well is the classification of low-dimensional semisimple Hopf superalgebras (or braided Hopf algebras) understood?

As far as I know, low-dimensional semisimple Hopf algebras are classified (along with non-semisimple ones) up to dimension 60, with the first example of a semisimple Hopf algebra not coming from a ...
0
votes
0answers
41 views

Is the projector to irreducible tensor modules of SO(N) known?

To project a generic tensor to an irreducible module of SO(N) one has to (anti)symmetrize the indices and then subtract traces, e.g. for symmetric traceless 2-tensors $$ \frac{1}{2} (\delta_{I_1 J_1} ...
0
votes
0answers
93 views

Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and every ...
8
votes
2answers
270 views

Sums of degrees of irreducible complex characters

The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant to determine the dimension of a maximal torus of the Lie algebra associated to ...
2
votes
1answer
123 views

Weight multiplicities for some particular representations of SO(2m).

I am looking for explicit formulas for the weight multiplicities of some particular irreducible representations of $SO(2m)$. It is possible that they have been already computed; in this case I will ...
2
votes
1answer
62 views

Pivotal functors of that are substantially different from finite group homomorphisms

Fusion categories can be seen as generalisations of the representation category of finite groups. I'm interested in spherical fusion categories. I'm trying to find "interesting" functors from a ...
2
votes
0answers
85 views

Is every irreducible unitary class one representation induced?

Let $G$ be a connected semi simple Lie group with finite center. Fix a maximal compact subgroup $K$. An irreducible representation $(\pi,V)$ of $G$ is called a "class-one representation", if it ...
2
votes
1answer
166 views

Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)

I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form $ \left( ...
6
votes
0answers
118 views

minimal energy of affine Lie algebra reps

Let $\mathfrak g$ be a simple Lie algebra. Let $\widetilde{L\mathfrak g}$ be the universal central extension of $L\mathfrak g:=\mathfrak g[t,t^{-1}]$. Let $V_\lambda$ be a positive energy ...
-1
votes
0answers
54 views

Complex conjugation of positive roots [migrated]

I have a simple question about root systems. Suppose that $G$ is a connected reductive group over the reals $\mathbb{R}$, and $T\subset G$ is a maximal torus (by this I mean that $T_{\mathbb{C}}$ is a ...
2
votes
1answer
186 views

Irreducible representation of Heisenberg group with characteristic 2?

As we all know that the irreducible representation for Heisenberg group can be classified easily when the group is over a finite field $\mathbb{F}_q$, where $q=p^n$ and $p$ is a prime greater than ...
1
vote
1answer
106 views

Representations of parabolic subgroups of the general linear group over the complex numbers

In all that follows, we are working over $\mathbb{C}$. Let $B \subseteq P \subseteq {\rm GL}(n)$ be a parabolic subgroup. Can you say anything in general about the representations of $P$? I suspect ...
1
vote
0answers
44 views

“embedding” various matrix equivalences into the equivalence of particular linear map

Consider the square matrices over a (local) ring $R$, up to conjugation, $A\rightarrow UAU^{-1}$, where $U$ is an invertible matrix over $R$. Such an equivalence embeds into the "left-right" ...
0
votes
1answer
76 views

Absolute irreducibility of a symmetric square?

This is a question I received today by email, which somebody more experienced with finite group representations can probably answer directly. Take $F:=\mathbb{F}_q$ for some prime power $q$, so ...
0
votes
0answers
63 views

reference help indecomposable representations of SL(2,R)

Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, $K=SO(2)$ the maximal compact subgroup of $SL_2(\mathbb{R})$. Then the classification of irreducible admissible ...
2
votes
1answer
176 views

What does the defect of a block measure?

In the context of decomposition matrices for Hecke algebras of finite Coxeter groups at a root of unity (such as the tables at the end of the book "Hecke algebras at a root of unity" by Geck-Jacon or ...
4
votes
2answers
116 views

Invariant planes of a nilpotent matrix with two Jordan blocks of size two

Describe all the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map $$ N = \begin{pmatrix} 0 & 1 & & \\ 0 & 0 & & \\ & & 0 ...
0
votes
0answers
174 views

PBW proof proposal

One version of the PBW theorem states: $\omega $:$\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras. I am curious if this is a possible proof for the PBW theorem, part is taken ...
3
votes
0answers
103 views

Non-linearly isomorphic non-equivalent $G-$representations?

Let $G$ be an algebraic group (or a group scheme) over a field $\Bbbk$, and let $V$ be an algebraic $G-$representation (I mean, corresponding to a homomorphism of $\Bbbk-$group schemes $G\rightarrow ...
0
votes
0answers
96 views

Arthur's refinement of parameters for unitary automorphic representations

In his work on the classification of automorphic representations of a group $G$, Arthur has conjectured that the parameterization of such representations involves a homomorphism $\rho : SL_2 \times ...
3
votes
1answer
258 views

An identity for elementary symmetric functions

Trying to understand a result in a representation theoretical paper, I realized that it implies the following elementary identity for symmetric functions. My question is whether this identity is true, ...
5
votes
3answers
373 views

Characterising the adjoint representation of SU(N)

One can show that the adjoint representation of $\mathrm{SU}(n)$, the image of the map $\mathrm{Ad}:\mathrm{SU}(n) \rightarrow \mathrm{Aut}(\mathrm{su(n)})\subset \mathrm{GL}(\mathrm{su}(n))$, is an ...
0
votes
1answer
113 views

Questions on invariant operators of finite group representations

1) Is there an equivalent of the Casimir operator for an irreducible representation of a finite group? 2) Given an invariant operator of a certain group, can I check if it is invariant under only ...
0
votes
0answers
73 views

Action of the (special) orthogonal group on differential forms

I was told that the following facts are true. I am looking for a reference to them. 1) The action of $O(n,\mathbb{C})$ on $\wedge^l\mathbb{C}^n$ is irreducible for any $l$. 2) The action of ...
2
votes
0answers
144 views

Highest weight spaces in arbitrary representations?

An isotypic (maybe reducible) representation V of GL(V) may be represented by its highest weight subspace HW(V). We have dim HW(V) equal to the multiplicity of the irreducible representation inside V ...
1
vote
0answers
102 views

Definition of 'Koszul Ring' (in BGS)

In the paper 'Koszul Duality Patterns in Representation Theory' by Beilinson et. al, they give the definition of a Koszul Ring: A Koszul ring is a positively graded ring $A = \bigoplus_{j \geq 0} ...
5
votes
1answer
168 views

Special linear groups contained in symplectic groups

Let $q$ be a power of prime $p$, and $n, m, k$ positive integers such that $mk=2n$ and $2\leq m<2n$. Let $\mathrm{Sp}(2n,q)$ be the symplectic group of dimension $2n$ over $\mathrm{GF}(q)$ and ...
6
votes
2answers
349 views

Quasi-affineness of the base of a $\mathbb{G}_a$-torsor

Let $\mathbb{G}_a$ be the additive group over an algebraically closed field $k$ of any characteristic. Let $X \to Y$ be a $\mathbb{G}_a$-torsor of $k$-schemes (of finite type - in case that is ...
0
votes
0answers
61 views

Determinant of an action and characters

In the paper of Ramanathan "Stable Principal Bundles on a Compact Riemann Surface", I read: ...where $\mu$ is the determinant of the (adjoint) action of $P$ on ...
6
votes
4answers
592 views

A question on non-archimedian Fourier transform

Let $M(n)$ be the vector space of $n\times n$ matrices over a local non-archimedian field $K$. Let $\mathcal S$ denote the space of locally constant compactly supported functions on $M(n)$. Similarly, ...
2
votes
0answers
59 views

nonvanishing of global theta lifting from U(1) to U(1?)

I understanding nonvanishing of theta lifting, either global or local, is a difficult and open problem. But I wanna know if there is an answer for the following simplest case. Let $E/F$ be a ...
1
vote
1answer
117 views

Successive Schur covers

Let $G_0$ be a finite group and $G_j$ a Schur cover of $G_{j-1}$ for $j=1,2,3\ldots$. Is $G_2$ equal to $G_1$? If not, will the sequence stop after finite steps in general?
18
votes
3answers
500 views

What's the state of affairs concerning the identification between quantum group reps at root of unity, and positive energy affine Lie algebra reps?

In his paper [1], Finkelberg used Kazhdan-Lusztig's massive work [4,5,6,7,8] to prove that $Rep^{ss}(U_q\mathfrak g)$ (the semisimplification of the category of finite dimensional reps of ...