All Questions
Tagged with reference-request rt.representation-theory
823 questions
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
4
answers
477
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 ...
- 1,595
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 ...
- 4,902
4
votes
3
answers
272
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 - e....
user27328
15
votes
2
answers
654
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 ...
- 21.8k
3
votes
2
answers
913
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, ...
- 2,709
6
votes
0
answers
3k
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
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 ...
1
vote
1
answer
308
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 complexification and $\tau: U^...
8
votes
3
answers
559
views
Reference for tetrahedral Coxeter group
Let $G$ be the group with 4 generators, each of order 2, such that the product of any 2, say $ab$, has order 3 (i.e., $ababab=e$).
That is, this is an infinite reflection group with Coxeter diagram a ...
- 83
6
votes
0
answers
134
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 $...
- 1,639
0
votes
0
answers
177
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 $\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?
- 393
9
votes
3
answers
894
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. ...
- 3,174
8
votes
2
answers
795
views
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 $GL(2,F)$ ...
- 5,562
7
votes
1
answer
672
views
The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group
I asked this question on Math.Stack but have not had any answers.
Question
What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group, $A$?
The trivial ...
- 1,037
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
10
votes
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
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 ...
- 6,201
12
votes
1
answer
2k
views
Wrong-way Frobenius reciprocity for finite groups representations
This is a typical lazy mathematician question, so do not hesitate to close it and recommend me to do my homeworks...
Let $H$ be a subgroup of a finite group $G$, and let $Res_H^G$ and $Ind_H^G$ the ...
- 6,777
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
4
votes
3
answers
2k
views
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 ...
- 247
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 ...
- 829
2
votes
1
answer
274
views
How to detect if a subgroup lands inside an orthogonal group?
Equivalently, my question may be phrased as, "Are there defining characteristics of representations of orthogonal (symmetric form-preserving) groups?"
Here I am working with a unitary representation ...
- 247
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-...
- 502
9
votes
2
answers
933
views
Good effective versions of theorems of Artin and Brauer
The theorem of Artin and Brauer of the title are the famous theorem in the theory of representation of finite groups.
For example, Artin's theorem is the statement that for every character $\chi$ of ...
- 26k
6
votes
2
answers
794
views
Plancherel formula for non-second-countable (non-unimodular) groups
The Plancherel formula for unimodular, second-countable, type 1 groups can be found in A Course in Abstract Harmonic Analysis by Gerald Folland (theorem 7.44) or here. It states that we can get a ...
- 2,442
11
votes
0
answers
870
views
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 ...
- 303
9
votes
5
answers
2k
views
A catalog of faithful representations of finite groups?
I want a reference that catalogs the smallest-dimensional faithful representation of every noteworthy finite group. Specifically, I want representations on $\mathbb{R}^n$ and $\mathbb{C}^n$.
Where ...
- 7,633
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 ...
- 52.9k
4
votes
1
answer
392
views
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 ...
- 4,902
0
votes
1
answer
315
views
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 ...
- 52.9k
5
votes
2
answers
347
views
Categorified versions of Mackey's functor
I would like to ask for possible references for the following very general situation, a categorified version of Mackey functors.
The question is if there are other known constructions to associate to ...
- 53
0
votes
0
answers
326
views
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 ...
- 829
21
votes
2
answers
944
views
Which p-adic algebraic groups are type I?
It was proved by Jacques Dixmier (Sur les représentations unitaires des groupes de Lie algébriques, Annales de l'institut Fourier, 7 (1957), p. 315-328, doi: 10.5802/aif.73, MR 20 #5820, Zbl 0080....
- 11.1k
6
votes
0
answers
549
views
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
4
votes
1
answer
869
views
Finite Unipotent Groups: References
It will be a great pleasure for me if one can suggest "Survey Articles" on following topics related to the finite unipotent group $U(n,\mathbb{F}_q)$. (Thanks in advance!!!)
The number of ...
- 1,169
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 ...
- 26k
3
votes
3
answers
1k
views
Generalization of Schur's Lemma
Let $\rho : G \to GL(V)$ be an irreducible representation of a finite group. Schur's lemma says if $\pi:GL(V) \to GL(V)$ intertwines with $\rho$, that is, $\pi \rho(g) = \rho(g) \pi$ for every $g\in G$...
- 651
10
votes
1
answer
2k
views
Permutation character of the symmetric group on subsets of certain size
The symmetric group $S_n$ acts on $[n]:=\{1,\ldots,n\}$, thereby inducing an action on the set $$\wp_k(n)=\{\: A\subseteq[n] \::\: \#A=k \:\}$$ of subsets of cardinality $k$, simply by $$(g,A)\mapsto ...
- 238
5
votes
2
answers
584
views
BGG-like resolutions and translations
This question originated from my confusion after I read the following paragraph (page 31, section 4.8) in Resolutions and Hilbert series of the unitary highest weight modules of the exceptional ...
- 8,597
8
votes
1
answer
400
views
Decomposition of $\mathrm{End}(V)$ as $S_n\times S_n$-module
Let $V$ be a finite-dimensional, complex vector space and set $\newcommand{\Gl}{\mathrm{Gl}}G:=\Gl(V)\times\Gl(V)$. Let $E:=\mathrm{End}(V)$ and consider its coordinate ring $\mathbb C[E]$, the space ...
- 4,902
2
votes
0
answers
239
views
Resolution of singularities of this cubic surface?
Let $A = \mathcal O(Y)^{SL_2(\mathbb C)}$ be the ring of invariant functions on $Y := \mathrm{Hom}(\mathbb Z^2, SL_2(\mathbb C))$. We can identify $A$ with the quotient of $\mathbb C[x,y,z]$ by the ...
- 2,869
1
vote
1
answer
322
views
translation functors in parabolic category $\mathcal{O}$
I'm looking for a reference for a treatment of translation functors (as defined e.g. in this [question][1]) in parabolic versions of BGG category $\mathcal{O}$.
I am mainly interested in the ...
- 8,597
19
votes
2
answers
1k
views
Reference request for Plancherel measure
I need a good reference for the basic definitions of the dual of locally compact group (not necessarily abelian), its natural topology, $\sigma$-algebra, and the Plancherel measure on it (when they ...
- 26k
3
votes
0
answers
209
views
What is known about 2-modular representations of Ree groups of type $F_4$?
A too-vaguely worded question posted today about Suzuki and Ree groups reminds me to revisit a concern I never followed up years ago when assembling information about modular representations of finite ...
- 52.9k
5
votes
2
answers
452
views
"geometric" description of the algebra of central functions on a Lie group
I am looking for a a description of the algebra of continuous central functions on a group, say a compact simple Lie group $G$, as the algebra of all continuous functions on a "nice" compact Hausdorff ...
- 2,201