All Questions
Tagged with reference-request rt.representation-theory
823 questions
6
votes
1
answer
255
views
Questions about the $\mathbf{i}$-trails of Berenstein and Zelevinsky
The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in this paper. It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^*$. Let ${\bf i}=(i_1, \...
- 6,201
11
votes
2
answers
2k
views
Representation theory of the general linear group over a finite prime field
I am re-posting a question I asked on math.se here because I am unsatisfied with the answers I obtained.
The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely ...
- 4,902
16
votes
1
answer
669
views
Subquotients in the Verma filtration on Verma modules
Let $\lambda$ be a dominant integral weight of $\mathfrak g$, a finite-dimensional reductive Lie algebra over $\mathbb C$. Let $M(w\cdot \lambda)$ denote the Verma module with high weight $w\cdot \...
- 27.8k
3
votes
0
answers
214
views
Unitary dual of $Sp_4(\mathbb{R})$
Do we know the unitary dual of $Sp_4(\mathbb{R})$? If so, can someone provide me any references? How about other rank 2 real groups? Thank you!
1
vote
0
answers
196
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 $A_6 \hookrightarrow PSU(3).$ ...
- 111
7
votes
1
answer
389
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 ...
- 1,942
3
votes
0
answers
168
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 ...
- 2,384
2
votes
1
answer
214
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
- 838
5
votes
0
answers
304
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 ...
- 1,942
3
votes
1
answer
113
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 zero....
- 1,545
5
votes
2
answers
312
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 one-...
- 27.8k
10
votes
1
answer
354
views
Derivation of Blattner's conjecture in the Beilinson-Bernstein picture
On the last page of Schmid's article "Discrete Series", he says
"In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb ...
- 27.8k
13
votes
3
answers
2k
views
on the center of a Lie group
I'm trying to set straight my various pieces of knowledge about the center of a compact Lie group, and I'm running in circles...
First some definitions:
• Let $G$ be compact, simple, and simply ...
- 43.2k
3
votes
0
answers
679
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) $\mathbb{C}...
5
votes
2
answers
760
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 semi-simple....
- 2,384
18
votes
5
answers
3k
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 ...
- 1,429
3
votes
2
answers
323
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 ...
- 5,565
8
votes
1
answer
308
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 ...
- 43.2k
0
votes
1
answer
186
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 $G:=\...
- 52.9k
4
votes
2
answers
341
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 $(\mathfrak{g},K)$-...
- 2,709
3
votes
1
answer
598
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, ...
- 21.8k
0
votes
0
answers
189
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 $SO(n,\...
- 21.8k
2
votes
0
answers
228
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 ...
- 893
6
votes
0
answers
225
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 two-...
- 17.4k
6
votes
0
answers
455
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 ...
- 1,719
10
votes
1
answer
377
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 ...
- 17.4k
3
votes
1
answer
462
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 ...
- 10.4k
6
votes
3
answers
757
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 $(SL_2(\mathbb{C}), PGL_2(O_F))$ (...
- 809
5
votes
1
answer
452
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 \frac{\...
- 1,906
4
votes
0
answers
176
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 ...
- 10.7k
3
votes
0
answers
220
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 ...
- 1,856
7
votes
0
answers
140
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 ...
- 1,980
3
votes
5
answers
1k
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 ...
- 2,508
5
votes
1
answer
404
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 $dg$...
- 2,554
4
votes
1
answer
256
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$. ...
- 4,902
25
votes
3
answers
6k
views
Introductory References for Geometric Representation Theory
Would anyone be able to recommend text books that give an introduction to Geometric Representation Theory and survey papers that give an outline of the work that has been done in the field? I'm ...
6
votes
0
answers
244
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)$, ...
- 27.8k
1
vote
0
answers
196
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 ...
- 111
4
votes
1
answer
592
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 ...
- 35.5k
5
votes
1
answer
216
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 and ...
- 5,654
3
votes
2
answers
319
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 ...
- 83
11
votes
3
answers
1k
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 ...
8
votes
1
answer
1k
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 ...
- 4,920
5
votes
1
answer
514
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]]$$ be its affine Grassmannian. I have read that $\mathcal{G}r$ possesses a natural very ample line ...
- 4,920
5
votes
2
answers
297
views
Decomposition into irreducibles of the representation $L^2(SL_2(\mathbb{C})/\Gamma)$ for $\Gamma$ geometrically finite
I am trying to understand the decomposition
$$L^2(SL_2(\mathbb{C})/\Gamma)=\oplus_i C_i \oplus V_{temp}$$
where $C_i$ are complementary series representations corresponding to eigenfunctions of the ...
- 51
4
votes
1
answer
321
views
Generalization of Frobenius formula involving Macdonald polynomials
Given a vector $\vec k=(k_1,k_2,\cdots)$ with $k_i$ are non-negative integers, the Newton polynomial $p_{\vec k}(x)$ is defined as
\begin{equation}
p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~,
\end{...
- 2,273
6
votes
1
answer
401
views
Stratifications and Filtrations of the Affine Grassmannian
Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r=G(\mathcal{\mathbb{C}((t))})/G(\mathcal{\mathbb{C}[[t]]})$$ be the affine Grassmannian of $G$. We know that $\...
- 4,920
7
votes
2
answers
1k
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 $\mathcal{G}$ ...
- 4,920
4
votes
3
answers
916
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 ...
- 4,902
12
votes
3
answers
2k
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 ...
- 5,565