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 ...
13
votes
3answers
628 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 ...
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 ...
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 ...
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, ...
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 ...
1
vote
0answers
96 views

References for crystal bases and Demazure modules in representation theory

I was wondering what are some standard general references/books/survey articles about: (1) crystal bases, and string parameterizations and (2) Demazure modules, and Schubert varieties (containing ...
4
votes
0answers
110 views

Parshin's buildings for higher local fields

What is the status of the theory of buildings for higher local fields? I know that there are some papers of Parshin, in which he describes some examples, like $PGL_2$ and $PGL_3$ over ...
4
votes
0answers
131 views

Cohomology of Bott-Samelson varieties?

How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here. Is there ...
9
votes
1answer
164 views

Fixed set of order p automorphism of Bruhat-Tits tree

I would like to know the structure of the fixed set of an order $p$ automorphism [Edit: induced by a matrix in $GL_2(K)$] on the Bruhat-Tits tree for a p-adic field $K$, specifically in the case where ...
2
votes
1answer
173 views

R-linear representations of sl(2,C)

Is there some good reference for the classification of finite-dimensional ${\mathbb R}$-linear (as opposed to ${\mathbb C}$-linear) representations of $\mathfrak{sl}_2{\mathbb C}$? Equivalently, what ...
5
votes
3answers
306 views

Decomposition of $L^2(\Gamma \backslash G)$

Let $G$ be a semisimple Lie group, and $\Gamma$ be an lattice (arithmetic) - typical examples I am thinking about would be $(SL_2(\mathbb{R}), SL_2(\mathbb{Z})$, or $(SL_2(\mathbb{C}), PGL_2(O_F))$ ...
3
votes
0answers
77 views

Orthogonal basis for the multilinear polynomials with zero “trace”

We say that a multilinear polynomial $P(x_1,\ldots,x_n)$ in $n$ commuting variables over $\mathbb{R}$ has zero trace if $$ \frac{d}{dt} P(t,\ldots,t) = 0. $$ Equivalently, $$ \left(\sum_{i=1}^n ...
0
votes
0answers
89 views

Reps of a compact connected Lie group are equivalent iff they are equivalent as reps of a maximal torus

Let $G$ be a compact connected Lie group, $T$ a maximal torus in $G$ and $V$, $W$ finite-dimensional $G$-representations. Using characters and the fact that every element of $G$ can be conjugated into ...
4
votes
0answers
139 views

Is there a notion of “tame” representations of $GL_n(Z)$?

This is a followup to this question about the (left) noetherianity of the group ring of $GL_n(\mathbf{Z})$: Does GL_n(Z) have a noetherian group ring? Given that $\mathbf{Z}[GL_n(\mathbf{Z})]$ is ...
3
votes
0answers
81 views

Shalika germ for local function field

I am wondering if there is a theorem of Shalika germ (as below) for local function field, for both the group version or the Lie algebra version, probably under assumption on the characteristic to be ...
8
votes
0answers
108 views

Algebraic construction of the modular representation of $\mathrm{SL}_2(\hat{\mathbf Z})$

The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if ...
1
vote
5answers
403 views

Character table of $S_7$

Is there any reference where I can find the character table of the symmetric group $S_7$? A simple search in google gave me a GAP program that computes the character table, but I don't understand the ...
5
votes
1answer
179 views

Equivariant Formality

Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$. Assume further there exists a ...
4
votes
1answer
98 views

Weyl group action on complexified Iwasawa decomposition

Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [SW02]. Let $T$ be a maximal torus of $B$, therefore a maximal torus of $G$. ...
6
votes
0answers
122 views

Zariski closure of orbits of real groups on complex flag manifolds

Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...
1
vote
0answers
47 views

Reference Help: Matsuki duality Orbits

I'm studying the Matsuki duality of $G_0$-orbits and $K$-orbits over a flag manifold $G/P$ where $G$ is semisimple complex Lie group and $P$ is a parabolic subgroup. I would like to study some ...
3
votes
1answer
330 views

What sort of ring-theoretic properties does the representation ring of a compact Lie group possess?

Recall the definition of the representation ring $R(G)$ of a compact Lie group $G$. I'd like a reference that gives me basic ring-theoretic properties that $R(G)$ always has, or enough info that I can ...
4
votes
1answer
153 views

To whom is the internal characterization of $Q$-groups due?

A group is said to be a $Q$-group if the character of any complex representation is rational valued. A well-known internal characterization of $Q$-groups is the following: $G$ is a $Q$-group if ...
2
votes
2answers
251 views

Density of characters

Did Harish-Chandra prove that characters of irreducible representations of a $p$-adic reductive group $G$ span a dense subspace of the space of conjugation-invariant distributions on $G$? What is the ...
10
votes
3answers
415 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 ...
2
votes
1answer
305 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 ...
4
votes
1answer
180 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 ...
5
votes
1answer
182 views

Decomposition into irreducibles of the representation $L^2(SL_2(\mathbb{C})/\Gamma)$ for $\Gamma$ geometrically finite

I am trying to understand the decomposition $$L^2(SL_2(\mathbb{C})/\Gamma)=\oplus_i C_i \oplus V_{temp}$$ where $C_i$ are complementary series representations corresponding to eigenfunctions of the ...
3
votes
0answers
132 views

Generalization of Frobenius formula involving Macdonald polynomials

Given a vector $\vec k=(k_1,k_2,\cdots)$ with $k_i$ are non-negative integers, the Newton polynomial $p_{\vec k}(x)$ is defined as \begin{equation} p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~, ...
5
votes
1answer
175 views

Stratifications and Filtrations of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r=G(\mathcal{\mathbb{C}((t))})/G(\mathcal{\mathbb{C}[[t]]})$$ be the affine Grassmannian of $G$. We know that ...
5
votes
2answers
321 views

Strata of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and denote by $\mathcal{G}$ its affine Grassmannian. Fix a maximal torus $T\subseteq G$. We know that $\mathcal{G}$ ...
4
votes
3answers
370 views

Clifford's Theorem with all its aspects in modern language, looking for a textbook

I am looking for a (more or less) introductory textbook on representation theory that contains the full contents of Clifford's paper "Representations Induced In An Invariant Subgroup" in modern ...
5
votes
3answers
420 views

What is a good introduction to branching rules in representation theory?

I'm looking for a book or introductory article, that explains branching rules in representation theory of real Lie groups. When a Lie group has a set of irreducible representations, I'd like to know ...
15
votes
0answers
269 views

Quasi-classical limit of representation theory

I am looking for a good reference on a general phenomenon of quasi-classical limit in representation theory, which relates "large" representations to measures on (co-adjoint orbits of) the associated ...
3
votes
3answers
292 views

Topological properties of $K$ orbits in $G/B$

I'll be working over the complex numbers. Let $G$ be a connected reductive group, $\theta\colon G\to G$ an involution. Let $K=G^{\theta}$ be the fixed point subgroup. I am trying to track down ...
8
votes
1answer
341 views

Action of the endomorphism monoid on an irreducible GL-module

Let $G=\mathrm{Gl}_n(\mathbb C)$ and $V$ an irreducible $G$-module on which $G$ acts polynomially. In other words, the algebraic group action of $G$ on the affine space $V$ extends to an algebraic ...
4
votes
3answers
230 views

Invariants in $S^n(S^k(\mathbb{C}^w)$

Are the formulas for the multiplicity of $SL(w)$ invariants in $S^n(S^k(\mathbb{C}^w)$ known? This is a very classical topic. If no, in what ranges one can compute it (for certain paramaters fixed - ...
10
votes
2answers
300 views

Invariant differential operators on real Grassmannians

I am looking for an explicit description of the algebra of $SO(n)$- or, better, $O(n)$-invariant differential operators on the real Grassmann manifolds of $k$-dimensional linear subspaces in the ...
0
votes
1answer
95 views

reference help: irreducible implies admissible

Let $G$ be a reductive p-adic group, $\pi$ a complex smooth representation of $G$. Then it is known that if $\pi$ is irreducible, then it is admissible. I need help to find a reference for this fact, ...
2
votes
0answers
257 views

Representations of the orthogonal group O(n) vs representations of the special orthogonal group SO(n), over an arbitrary field

Let $O(n)$ and $SO(n)$ denote the split orthogonal linear algebraic group and its special subgroup, over some fixed field of characteristic not two. I am looking for a reference that explains how to ...
3
votes
2answers
248 views

Simple representations of products of algebraic groups

I am looking for a reference for the following assertion that I believe to be true. All representations are assumed to be finite-dimensional. Let $G_1$ and $G_2$ be affine algebraic group schemes ...
1
vote
1answer
199 views

Holomorphic representations of complex reductive Lie groups and the boundary of orbits (Reference request)

I have difficulties finding an appropriate reference for the following question (which I hope that it to be true). Let $U$ be a compact Lie group, $G:=U^{\mathbb{C}}$ its complexi´Čücation and $\tau: ...
3
votes
0answers
64 views

Do purification and equivariantization commute?

Suppose that we have an action of a group $G$ on a (quasi-)Hopf algebra $H$, so that we can construct $H\rtimes G$ as in Majid's Cross Products by Braided Groups and Bosonization. It is known that ...
0
votes
0answers
72 views

Existence of special pants decompositions for non-elementary representations into PSL(2,R)

A Theorem by Gallo, Goldman and Porter states the following: Let $S_g$ be a closed orientable surface of genus $g$ with fundamental group $\Gamma_g$, and fix a non-elementary representation ...
8
votes
2answers
541 views

Isomorphism between Spin(3,2) and Sp(4, R)

I've been using the fact that Spin(3,2) is isomorphic to Sp(4, R) for a while, but I've never seen a proof. Can anyone point me in the direction of a good reference?
6
votes
3answers
322 views

Representation rings of exceptional Lie groups

Let $G$ be a compact Lie group and let $R(G)$ denote its complex representation ring. If $G$ is simply connected, such as $G_2$, $F_4$ or $E_8$, then it is known that $R(G)$ is a polynomial ring [F. ...