All Questions
Tagged with rt.representation-theory reference-request
823 questions
6
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0
answers
244
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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
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196
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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
392
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Expression of basis vectors of permutation modules in different bases.
This is a cross-post from math.se, because I did not get any answer there:
Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that ...
CommunityBot
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16
votes
2
answers
992
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Maximal number of maximal subgroups
Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...
4
votes
1
answer
592
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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 ...
- 118k
5
votes
1
answer
216
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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 ...
CommunityBot
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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 ...
- 6,406
5
votes
1
answer
514
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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 ...
- 171
8
votes
1
answer
1k
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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 ...
- 27.8k
5
votes
2
answers
3k
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irreducible representations of O(2) - reference?
I am looking for a list of the irreducible representations of O(2). Could someone please provide a reference?
EDIT: I am particularly interested in the representations on IR^2 (irreducible or not)
- 11.2k
4
votes
3
answers
916
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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 ...
- 44.4k
18
votes
0
answers
469
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 ...
- 1,765
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 $\...
- 2,554
26
votes
1
answer
2k
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Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?
This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of ...
CommunityBot
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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 $\mathcal{G}$ ...
- 2,554
12
votes
3
answers
2k
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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 ...
- 23k
8
votes
1
answer
382
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 ...
- 2,649
4
votes
3
answers
272
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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 - e....
- 26.3k
4
votes
2
answers
824
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decomposition into irreducible unitary representations: references for explicit formulas?
I'm looking for references of the decomposition of $L^2(\Gamma\backslash G)$, where $G$ is a connected Lie group, and $\Gamma\subset G$ a discrete lattice; for simplicity one may assume that $G$ is ...
- 11.2k
1
vote
1
answer
308
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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 complexification and $\tau: U^...
CommunityBot
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18
votes
2
answers
2k
views
Virasoro action on the elliptic cohomology
I'm trying to understand better the mathematical notion of elliptic cohomology. Note that I only know the physics definition of the elliptic genus given in Witten's paper.
Let $X$ be a Calabi-Yau ...
- 5,186
6
votes
0
answers
3k
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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 ...
CommunityBot
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3
votes
2
answers
654
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 ...
12
votes
2
answers
3k
views
Is Lusztig's conjecture solved?
What I said is Lusztig's conjecture about representation of quantum group at root of unity and representation of Lie algebra at positive characters.
It seems that Andersen-Jantzen-Soergel ever wrote ...
- 44.7k
6
votes
3
answers
650
views
Orthogonal subgroups of dual group
This question arises from this one.
Let $G$ be a finite abelian group and $H$ a subgroup of $G$. Let $\widehat{G}$ be group of all characters of $G$ and let $H^\perp = \{\chi \in \widehat{G} : \chi ...
CommunityBot
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6
votes
0
answers
134
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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 $...
- 1,639
0
votes
0
answers
177
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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 $\rho\...
- 3,749
15
votes
2
answers
2k
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?
- 45.7k
8
votes
2
answers
795
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Proving that some principal series representations of SL(2,F) are irreducible
I am sorry in advance if this question is not "research level".
Let $F$ be a p-adic field.
I saw, in Bumps book, a proof which I liked, showing which principal series representations of $GL(2,F)$ ...
- 11.2k
10
votes
3
answers
895
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. ...
- 5,191
10
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0
answers
430
views
A question about multiplication in $G(\mathbb{C}((t)))$ and Affine Grassmannians
I am sorry to give a bounty to such a crappy question but an answer would help me a lot.
I am stuck with the following simple (i guess but) technical problem.
Let $G$ be a complex reductive ...
- 2,554
5
votes
0
answers
324
views
A question about equivariant derived categories and [BBD]
Let $G$ be an algebraic group (over $\mathbb{C}$) acting algebraically on a variety $X$. Bernstein and Lunts then define in [BL94] the equivariant derived category $D^b_G(X,\mathbb{C})$ (of $\mathbb{C}...
- 2,554
2
votes
2
answers
220
views
References request: representations of Heisenberg algebra.
Let $p_1, p_2, \ldots$, be the power sum symmetric functions. Let $p_n^* = n \frac{\partial}{\partial p_n}$. Then $$ p_n^* p_m - p_m p_n^* = \delta_{m, n} 1. $$
Where could I find this result in some ...
- 11.2k
4
votes
3
answers
2k
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Decomposition into irreducibles of symmetric powers of irreps.
Suppose I have an irreducible representation of a simple Lie algebra, say $\mathfrak{sl}(n)$ or $\mathfrak{so}(n)$ i.e., $A$ and $D$ type. Given such a representation, $\Gamma_\lambda$, indexed by its ...
11
votes
2
answers
606
views
Temperley-Lieb algebras for other Weyl groups?
The Temperley-Lieb algebra has the same generators as the $S_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen ...
- 27.8k
7
votes
1
answer
865
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Associated vector bundles of infinite rank and induced connections
Let $\mathbb{V}$ be a representation of a Lie group $G$ and let $P \to M$ be a principal $G$-bundle with a principal connection. If $\mathbb{V}$ is finite-dimensional, then one can associate to this ...
- 25.3k
13
votes
0
answers
1k
views
Source of a formula for tensor product multiplicities?
This is a follow-up to a recent question by Allen Knutson here, involving a special type of tensor product multiplicity for a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (or other ...
CommunityBot
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1
vote
1
answer
362
views
Free resolution for Lie algebras (reference)
What is a reference for the subject of "free resolutions for Lie algebras"?
Does the term "standard resolutions" means "free resolutions"?
What is a "bar resolution"?
Is there only one way to talk ...
CommunityBot
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2
votes
1
answer
169
views
Reference request : dimensions of real representations of Lie groups
What is a good reference to learn about real representations of Lie groups ? I've parsed through the very enlightening book of Fulton and Harris, but it is extremely (if not exclusively) example-...
- 25.3k
11
votes
0
answers
870
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Reference/quote request: "All of combinatorics is the representation theory of $S_n$"
I think I remember reading somewhere a glib (or is it deep?) quote, perhaps due to Rota?, which was something like
"All of combinatorics is essentially [or can be reduced to?] the representation ...
- 3,230
0
votes
1
answer
152
views
Continuation of homomorphisms of representations...
Hi all.
If $G$ is a finite group and $\varrho : G \to \text{GL}(V), \eta : G \to \text{GL}(W)$
are finite dimensional representations, $V_0$ is a $G$-invariant subspace of $V$
and $f : V_0 \to W$ is ...
- 3,545
5
votes
3
answers
330
views
reference for list of left-regular representations of real associative algebras
Suppose $\mathcal{A}$ is a unital associative algebra over $\mathbb{R}$. If we identify $\mathcal{A} = \mathbb{R}^n$ then the $\mathcal{A}$ multiplication corresponds to particular linear maps on $\...
- 2,804
0
votes
1
answer
315
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Orbital integrals of pseudo coefficients of supercuspidal reps
Let $\pi$ be a supercuspidal representation of $G =GL_2(F)$ for a non-archimedean local field $F$, then there exists a maximal subgroup $K$ of $G$, which is compact modulo the center, and a ...
- 11.2k
4
votes
1
answer
713
views
Criterion for nilradical of a maximal parabolic subalgebra to be abelian?
This question has some overlap with previous ones but doesn't seem to have a well-documented answer. I recall some literature (mostly involving Lie groups and hermitian symmetric pairs, etc.) which ...
CommunityBot
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0
votes
0
answers
326
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Endomorphism ring of a direct sum of tilting modules
I have found that a category of modules over a Lie algebra has an infinite number of (partial) tilting modules and that direct sum of these tilting modules is also an object in this category.
What ...
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6
votes
0
answers
549
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Categorical Koszul Duality as a form of geometric Langlands
I hope this question is not too unspecific:
Can Soergel's Categorical Local Langlands conjectures [1]
be interpreted as special form of geometric Langlands.
I think this is somehow hidden in the ...
- 2,554
3
votes
0
answers
163
views
reference request about fact the character of irreducible representation determine the representation itself.
It is well known that if two (irreducible) admissible representations have the same characters, then they are isomorphic. To my knowledge, this is true for both Lie groups and p-adic groups.
In the ...
- 2,709
3
votes
1
answer
243
views
Spectral synthesis for central functions on locally compact groups
There is a large literature on harmonic analysis on locally compact group, that
I am just beginning to discover. However I have not seen so far anything that emphasizes the central functions on $G$. A ...
CommunityBot
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5
votes
2
answers
923
views
Status of a conjectural definition of H. Nakajima
In his paper '$t$-analogue of $q$-characters of finite dimensional representations of quantum affine algebras' - http://arxiv.org/abs/math/0009231 - H. Nakajima states a conjectural definition of the $...
- 3,051
31
votes
1
answer
5k
views
Modern proof of Serre's open image theorem?
Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic Curves'...
- 1,905