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

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Representation theory of simple Lie algebras

Let $V$ be a simple complex Lie algebra. Let $W=\Lambda^2V$ be the second exterior power of $V$. Is it possible to find a basis for $W$ that consists of elements of the form $v \wedge w$, where $v$ ...
2
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2answers
228 views

Generalizations of Lie algebras

I often stumble over the term "Lie superalgebra" (= "Lie algebra with a $\mathbb{Z}_2$ grading"). Obvious question: What about $\mathbb{Z}_3$ grading (and so on)? Is a Lie algebra with $\mathbb{Z}_n$ ...
6
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1answer
228 views

Can Galois conjugates of lattices in SL(2,R) be discrete?

Let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$. Suppose that the trace field of $\Gamma$ is a totally real number field of degree $d$. This gives $d$ homomorphisms $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...
4
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2answers
327 views

Irreducible representations of compact groups

Let G be a compact group (or even profinite - Galois group). Let $V$ be a vector space over the field ${\mathbb F}_p$ with $p$ elements, $p$ a finite prime, such that $V$ is a contable product of ...
11
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1answer
356 views

Can one explain Tannaka-Krein duality for a finite-group to … a computer ? (How to make input for reconstruction to be finite datum?)

Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...
2
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1answer
213 views

Grothendieck group of representations

For a linearly reductive group $G$ over $k$ we consider the bounded derived category of finite dimensional representations $D^b(\mathrm{Repr}(G))$. Is the Grothendick group $K_0(D^b(\mathrm{Repr}(G))$ ...
0
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1answer
160 views

number of simple representations

For a linearly reductive Group $G$ over a field $k$ one has that the category of finite dimensional representations of $G$ is semisimple. What can one say about the number of simple representations? ...
0
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1answer
111 views

Is the invariant algebra semisimple

Let $A$ be a semisimple $k$ algebra and $k$ is of characteristic zero. Let $G$ be a linearly reductive group over $k$ acting on $A$. Is $A^G$ semisimple ?
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global dimension II

Suppose we are given finite dimensional semisimple $k$-algebras $A_1,..., A_r$. now we consider the matrix algebra$A=\begin{pmatrix} A_1 & M_{1,2} & \dots & M_{1,r} \\ 0 & A_2 ...
4
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2answers
327 views

Character values bounded away from zero

Character values for a finite group are sums of nth roots of unity. I'm wondering if there are any results bounding nonzero values of irreducible characters away from zero. Or if not are there ...
1
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0answers
127 views

Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field

Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field. Then $G(F)$ is a p-adic group. Let $\Psi(G)$ be the lattice of algebraic characters. Let $\Lambda_G$ be the ...
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402 views

Resource for learning quantum mechanics from the viewpoint of representation theory

Quantum mechanics is deeply connected with representation theory. Therefore, I'm looking for a textbook or article which presents quantum mechanics in a representation theoretic manner. Could anyone ...
1
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1answer
133 views

adjoint action of a Levi subalgebra

We work over an algebraically closed field of characteristic 0. Let $\mathfrak{g}$ be a reductive Lie algebra and let $\mathfrak{p}\supset\mathfrak{m}$ be a parabolic subalgebra, respectively a Levi ...
5
votes
1answer
169 views

global dimension

Suppose we are given finite dimensional semisimple $k$-algebras $A_1,..., A_r$. now we consider the matrix algebra$A=\begin{pmatrix} A_1 & M_{1,2} & \dots & M_{1,r} \\ 0 & A_2 ...
6
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3answers
312 views

Representations of S_n induced from centralizers of elements

Does anyone have a reference for a good description of representations of $S_{n}$ obtained by inducing up from $C_{S_{n}}(\pi)$, for some element $\pi$ of $S_{n}$? (I'd prefer an efficient ...
6
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1answer
323 views

Origin of symbols used for half-sum of positive roots in Lie theory?

The Weyl character formula is a central result in the finite dimensional representation theory of semisimple Lie groups, algebraic groups, Lie algebras. Related questions on MO include these here ...
5
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1answer
386 views

Origin of the term “weight” in representation theory

In representation theory, there are the related concepts of weights and roots. Since both are kinds of generalised eigenvalues, and eigenvalues are roots of e.g. the characteristic polynomial, the ...
12
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3answers
1k views

History and motivation for Tannaka, Krein, Grothendieck, Deligne et al. works on Tannaka-Krein theory?

I am trying to wrap my mind around Tannaka-Krein duality and it seems quite mysterious for me, as well, as its history. So let me ask: Question: What was the motivation and historical context for ...
3
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0answers
141 views

Finding a basis for the (linear combinations) span of a matrix group, efficiently?

I have an algorithm whose bottleneck is the following task: Let $\mathbb{F}$ be a finite field. Given a set of $k$ invertible matrices $g_1,\dots,g_k\in GL_n(\mathbb{F})$, let $G=\langle ...
9
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4answers
404 views

When are those subgroups of $\mathrm{SL}(2, \mathbb{C})$ discrete?

Let $A = \pmatrix{1 & 0 \\ \alpha & 1} $ and $ B = \pmatrix{1 & 1 \\ 0 & 1}$, where $\alpha \in \mathbb{C}$ is a complex parameter. Now consider the family of representations ...
3
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0answers
88 views

Enumerating simple algebraic groups and their irreducible representations

Motivation Everything is over an algebraically closed field. Given a faithful representation $G \to \textrm{GL}(V)$, one may try to pin down what the group $G$ exactly is (e.g., the in the case of ...
12
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2answers
471 views

Etymology of cuspidal representations

In the literature on representation theory of $GL_2(\Bbb F_p)$ and $GL_2(\Bbb Q_p)$, the irreducible representations with trivial Jacquet module are often called "cuspidal" or "supercuspidal". Why are ...
6
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2answers
240 views

Whitney stratification and affine grassmanian

Let $G$ a simply connected group over $\mathbb{C}$ and $Gr:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ the affine grassmannian. By Cartan decomposition we have a partition of stratas indexed by ...
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0answers
131 views

Canonical basis of quantum groups

I am trying to understand the canonical basis of quantum groups and different ways to construct the canonical basis of quantum groups. In the comments of Lusztig's papers, the paper [92], CANONICAL ...
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117 views

Restriction of the Steinberg representation

Let $G$ the group $GL(n,F)$ where F is a locally compact non Archimedean field, and $G^{0}$ the subgroup of $G$ consists of elements $g$ in $G$ such that $\det(g)$ in $\mathcal{O}_{F}^{\times}$, where ...
2
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1answer
92 views

Is is possible to lift an equivariant map of Loop lie algebras to an equivariant map of Loop groups?

For brevity, let $LG=\mathbb{T}\ltimes \tilde{L}G$, the affine loop group and let $G$ be a simple simply conneceted Lie group. I have a map $\phi:L\mathfrak{g} \to L\mathfrak{g}$ that is equivariant. ...
5
votes
2answers
333 views

Decomposing the conjugacy representation of Sym$(n)$ for small $n$

I am trying to compute the decomposition of the conjugacy representation of some small symmetric groups. Perhaps someone has undertaken a similar calculation. My own calculations are quite slow, ...
25
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5answers
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Why we need to study representations of matrix groups?

Why we need to study representations of matrix groups? For example, the group $SL_2(F_q)$, where $F_q$ is the field with $q$ elements, is studied by Drinfeld. I think that these groups are already ...
16
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1answer
611 views

Examples of finite groups with “good” bijection(s) between conjugacy classes and irreducible representations?

For symmetric group conjugacy classes and irreducible representation both are parametrized by Young diagramms, so there is a kind of "good" bijection between the two sets. For general finite groups ...
5
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0answers
131 views

A question on the resolution of parabolic Verma module $M_I(\lambda)$ in BGG category O

I am reading Humphrey's book "Representations of semisimple Lie algebras in the BGG Category O" on Page 189, Proposition 9.6, where he remarked that "Note that if we had developed the full BGG ...
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radical unipotent of a parahoric

Let $G$ a split connected reductive group over $\mathbb{C}$. $F=\mathbb{C}((t))$ and $\mathcal{O}$ the ring of integers. Let $B$ a Borel subgroup and $I$ the corresponding Iwahori. Let ...
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4answers
437 views

Structure of the adjoint representation of a (finite) group (Hopf algebra) ?

Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation. Question 1: what is known about this representation ...
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85 views

intersection of Borel in wonderful compactification

Let $\overline{G}$ the wonderful compactification of an adjoint $G$ over an algebraically closed field $k$. Let $B$ a Borel of $G$ and $w\in W$ an element of the Weyl group and $\overline{B}$ the ...
2
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1answer
296 views

The Bialynicki-Birula Stratification of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple group with affine Grassmannian $\mathcal{G}r$. Fix a maximal torus and Borel $T\subseteq B\subseteq G$. I am reading "Loop Grassmannian ...
2
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1answer
241 views

Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$

Let $\lambda_1, \dots, \lambda_n$ be the eigenvalues of a random Unitary matrix. I am interested in the expected value: $$\mathbb{E}_{X \in U(N)}\left[ \prod_{i=1}^n \frac{1}{(1 - ...
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5answers
665 views

Are there any known criteria for quadratic mapping from R^n to R^n being surjective?

Let quadratic mapping be the function from $\mathbb{R}^n$ to $\mathbb{R}^n$, where each coordinate is a quadratic form of $n$ variables. Are there any known criteria for it being surjective? May ...
5
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2answers
276 views

Relations between affine Grassmannian and Grassmannian

Let $\mathcal K = k((t))$ be the field of formal Laurent series over $k$, and by $\mathcal O = k[[t]]$ the ring of formal power series over $k$. Let $G$ be an algebraic group over $k$. The affine ...
4
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1answer
178 views

Reference for the Natural Ample Line Bundle on the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r:=G((t))/G[[t]]$$ be its affine Grassmannian. I have read that $\mathcal{G}r$ possesses a natural very ample line ...
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0answers
146 views

Derived category of representations

Suppose we are given an algebraic group $G$ (linearly reductive). Let $D^b(Repr(G))$ be the bounded derived category of finite dimensional algebraic representations of $G$. I am intressted in tilting ...
4
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1answer
216 views

The existence of a finite dimensional Lie algebra with a given symmetric invariant metric

The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...
4
votes
1answer
218 views

Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$

Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...
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0answers
204 views

Commutators for quantum Lie algebras

Can the usual definition of a Lie algebra via commutators be simply adapted to quantum Lie algebras? Graphically you have the IHX scheme, with the X being a virtual crossing (so to say). Does it ...
3
votes
2answers
326 views

Complete classification of six dimensional non-semi simple Lie algebra

I would aim to know the complete classification of 6 dimensional non-semi simple Lie algebra (here the dimension stands for the generators; or the dimension $\leq 6$). In this paper, in page 7, it ...
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1answer
138 views

Highest weights of irreducible components of tensor product of irreducible sl(3)-module [closed]

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows: For each weight $\mu$, let $L(\mu)$ be the irreducible ...
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2answers
257 views

Decomposing representations of GL(n,F_q) induced from certain kinds of parabolics

The answer to the question below is almost certainly known to the representation theorists; in fact, I'm pretty sure it can be extracted from Green's paper "The characters of the finite general linear ...
2
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1answer
191 views

Question about a proof in Graham and Lehrer's “Cellular algebras”

I'm sorry if this question is too basic for MO. I'm reading a paper by Graham and Lehrer "Cellular algebras" and have trouble understanding one step in a proof of a crucial theorem. I suppose that the ...
3
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3answers
411 views

Definition of Hecke operators

I am confused about the definition of Hecke operators. It will be great if someone provides some references. Shimura's 'Arithmetic Theory of Automorphic forms' says: Let $\Gamma$ be acting in the ...
4
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3answers
572 views

Reg the motivation behind Lusztig-Vogan bijection

Let $G$ be an algebraic group. Choose a Borel subgroup $B$ and a maximal Torus $T \subset B$. Let $\Lambda$ be the set of weights wrt $T$ and let $\mathfrak{g}$ be the lie algebra of $G$. Now, ...
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1answer
101 views

Fell topology in terms of distributions

Question: Can the Fell topology be expressed in terms of the distributions of the the tracial states of a unitary representations, that, is $\pi_j \rightarrow \pi$ if and only if $tr\; \pi_j ...
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3answers
241 views

Finding a character of height zero

My character theory is rather weak, so excuse me if this is a triviality. I have read on the encyclopedia of maths that for any group $G$, every block of $G$ contains an irreducible character of ...