**3**

votes

**1**answer

235 views

### 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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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 ?

**1**

vote

**0**answers

96 views

### 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**

votes

**2**answers

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**

vote

**0**answers

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 ...

**10**

votes

**3**answers

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**

vote

**1**answer

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

**1**answer

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**

votes

**3**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**3**answers

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**

votes

**0**answers

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**

votes

**4**answers

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**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**2**answers

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 ...

**2**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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

**2**answers

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**

votes

**5**answers

2k views

### 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**

votes

**1**answer

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**

votes

**0**answers

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 ...

**2**

votes

**0**answers

70 views

### 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 ...

**4**

votes

**4**answers

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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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 - ...

**14**

votes

**5**answers

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**

votes

**2**answers

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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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

**1**answer

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 ...

**2**

votes

**0**answers

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

**2**answers

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 ...

**0**

votes

**1**answer

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 ...

**6**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**3**answers

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**

votes

**3**answers

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, ...

**1**

vote

**1**answer

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 ...

**3**

votes

**3**answers

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 ...