All Questions
Tagged with reference-request rt.representation-theory
269 questions with no upvoted or accepted answers
18
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0
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612
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Who first noticed the duality for finite groups?
A.A.Kirillov in section 12.3 of his "Elements of the Theory of Representations" writes that the first "symmetric" duality theory for non-commutative groups was the theory for finite groups. In short ...
- 7,426
18
votes
0
answers
469
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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
18
votes
0
answers
895
views
local equivalence of loop group representations
Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the local loop group
$$
L_IG ...
- 43.2k
16
votes
0
answers
188
views
Representation theory of Pin groups
I am (still) thinking about branching rules from $\mathfrak{so}(n+m)$ to $\mathfrak{so}(n) \oplus \mathfrak{so}(m)$, using Proctor's paper as the starting point.
Proctor describes this rule for $m = 2$...
- 1,574
14
votes
0
answers
298
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Representation theory of Kac-Moody algebras in positive characteristic
I have been trying to learn about the representation theory of Kac-Moody algebras in positive characteristic. However, my usual reference for the subject (Infinite dimensional Lie algebras by Kac) ...
- 1,389
13
votes
0
answers
237
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A Dynkin type classification result in linear algebra
Let $G$ be a finite directed acyclic graph. The Cartan matrix $C_G=C$ of $G$ is defined as the matrix with rows and colums indexed by the vertices of $G$ and $c_{i,j}$ counts the number of paths from $...
- 26.5k
13
votes
0
answers
523
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Euler Subgroups and Automorphic L-functions
Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...
- 2,516
13
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0
answers
1k
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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
13
votes
0
answers
615
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The derived category of integral representations of a Dynkin quiver
Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write $\...
- 3,174
12
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402
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Which abelian categories have homological dimension 1?
In this MSRI lecture Geometry of Quiver Varieties I, Victor Ginzburg describes all abelian categories of homological dimension $1$ as being either
a category of representations $\mathrm{Rep}_\mathbf{...
- 1,161
11
votes
0
answers
818
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How to compute Ext-groups for categories without enough injectives/projectives?
I am studying a category of representations of an infinite-dimensional Lie algebra, which is an abelian category. Unfortunately, the category does not have enough injectives/projectives. I would ...
- 507
11
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0
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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
10
votes
0
answers
234
views
Branching from GL(a+b) to GL(a) x GL(b)$ using Gel'fand-Cetlin patterns
If one iterates the multiplicity-free branching rule from $\operatorname{GL}(n)$ representations (finite-dim, over $\mathbb C$) to $\operatorname{GL}(n-1)$ all the way down to $\operatorname{GL}(0)$, ...
- 27.8k
10
votes
0
answers
430
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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
10
votes
0
answers
881
views
Invariance of Euler characteristic under base change for sheaf cohomology of flag varieties
BACKGROUND:
Over an algebraically closed field of arbitrary characteristic, most of the basic structure theory of affine (= linear) algebraic groups can be developed concretely without quoting ...
- 52.9k
10
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0
answers
1k
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Complexes of representations with complementary central charges
This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary ...
- 15.6k
9
votes
0
answers
254
views
An identity for characters of the symmetric group
I am looking for a reference for the identity
$$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$
for the irreducible characters of the ...
- 2,306
9
votes
0
answers
286
views
Is Landvogt's thesis "The functorial properties of the Bruhat–Tits building" available online?
Universität Münster publishes theses online through "miami", but "miami" doesn't have Erasmus Landvogt's thesis (search).
ProQuest (predatorily) provides many theses, but they don'...
- 12.9k
9
votes
0
answers
210
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Why and how is a representation "continuously decomposable"?
What I am asking may apply to a much more general setting and I am interested in the underlying level of generality of these statements, mostly with canonical references. However I state the question ...
- 5,424
9
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0
answers
470
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Branching rules for compact Lie groups
Let $G$ be a compact connected Lie group, and let $H\subset G$ be a closed subgroup. For an irreducible representation $\pi:G\to\mathrm{End}_\mathbb{C}(V)$ of $G$ ($\dim\pi<\infty$) I want to know ...
- 1,959
9
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409
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The proof of Kazhdan's density theorem (And does it hold over positive characteristic?)
When proving identities about traces of functions on representations of $p$-adic groups, Kazhdan's density theorem indicates one only has to check equalities of traces on tempered representations. ...
- 181
8
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0
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481
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Formal degree of discrete series representations
Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is said to be a discrete series if its matrix coefficients (defined by $\xi^\pi_{v,w} : g \...
- 5,424
8
votes
0
answers
222
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references on categorification of knot invariants
I am extremely sorry if this is not the right place for this kind of question.
I have studied some knot theory, quantum invariants and would like to study more about categorification of knot ...
- 81
8
votes
0
answers
545
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What are the character tables of the finite unitary groups?
I need to know the (complex) character table of the finite unitary group $U_n(q)$. Lusztig and Srinivasan (1977) provide an abstract description, but parsing it requires a stronger background in ...
- 7,633
8
votes
0
answers
411
views
Which representations of the Lie algebra of a Lie group come from representations of the group itself?
I think this is very classic mathematics, but I can't find a complete answer in the literature.
Let $G$ be a Lie group, $\mathfrak{g}$ the Lie algebra of $\mathfrak{g}$. Suppose $\rho : \mathfrak{g} \...
- 1,586
8
votes
0
answers
129
views
Is there a splitting rule for the restriction of a $GL(23, \mathbb{Q})$-representation to $O(23, \mathbb{Q})$?
I am interested in a $23$-dimensional $\mathbb{Q}$-vector space $V$ which I am viewing as a GL$_{23}(\mathbb{Q})$ representation. Schur functors can be defined over $\mathbb{Q}$, so we get ...
- 233
8
votes
1
answer
849
views
Representations of groups with the same derived group, how much control do we have over the central character?
Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, ...
- 6,180
8
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0
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134
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Rational homotopy type of Hilbert scheme components
What is known about rational homotopy type of irreducible component of $Hilb^n(\mathbb C^k)$ containing configuration space? I've searched arXiv for a while and found nothing but surmise that Betti ...
- 4,600
8
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0
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370
views
When is a $2$-Calabi–Yau triangulated category the cluster category of a QP?
Keller–Reiten's main theorem in Acyclic Calabi–Yau categories implies that if $\mathcal{C}$ is a $2$-Calabi–Yau (algebraic) triangulated category admitting a cluster-tilting object $T$ such that the ...
- 1,690
8
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0
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388
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Reference Request - Spaces of Smooth Vectors
I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...
- 431
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
7
votes
0
answers
141
views
Frenkel-Kac's vertex operator realisation of the basic representation of an untwisted affine Kac-Moody algebra
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}} := (\mathfrak{g}[t^{\pm 1}] \oplus \mathbb{C} c) \rtimes \mathbb{C} D$ be the corresponding ...
- 1,516
7
votes
0
answers
176
views
The quotient of a higher Specht polynomial over the corresponding regular Specht polynomial
I'll need some notation before I can phrase my question, so please bare with me for a little. I'll try to get there as fast as possible (it's also my first MO question...).
Let $\lambda$ be a ...
- 131
7
votes
0
answers
207
views
A reference for Bernstein's approach to KL conjectures
The exposition in the famous J. Bernstein's lecture notes on D-modules is rather sketchy.
Fortunately, there are many other sources for the information from this text (like Hotta's excellent "D-...
- 81
7
votes
0
answers
171
views
$\mathfrak{sl}_2(\mathbb{Z})$'s properties as a Lie algebra over a ring
I was wondering what was known about $\mathfrak{sl}_2(\mathbb{Z})$ as a Lie algebra; in particular, what is known about its representation theory? I know of some texts which treat Lie algebras ...
- 71
7
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0
answers
107
views
Reference request: superconformal algebras and representations
I am looking for a book/monograph which deals with superconformal (vertex operator) algebras and their representation theory. What are some good books to understand to begin with the definition of a ...
- 171
7
votes
0
answers
202
views
What is the kernel of the action of the Iwahori-Hecke algebra?
The Iwahori-Hecke algebra $H_n(q)$ acts on the $n$th tensor power of the standard representation of $U_q(\mathfrak{sl}_m)$. What is the kernel of this action? Does anyone know a reference?
I'm happy ...
- 425
7
votes
0
answers
597
views
Reference for shtuka and trace formula
I really want to learn the work of Laurent Lafforgue and the joint work of Zhiwei Yun and Wei Zhang. They both involve shtuka and trace formula, which I only know the basic idea. So I would like to ...
- 409
7
votes
0
answers
543
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Representation theory of symmetric group for dummies
I have to grade a master project on representations of symmetric groups (char $0$) third time in my life and finally I came to a conclusion that I have to get a grasp of the matter. I am aware of ...
- 14.3k
7
votes
0
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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
6
votes
0
answers
236
views
Group homomorphism from $\mathrm{GL}_p$ to $\mathrm{SL}_p$ in characteristic $p$
If $k$ is a commutative field of characteristic $p>0$, then the map
$$ \theta \colon \mathrm{GL}_p(k) \to \mathrm{SL}_p(k) \colon A = (a_{ij}) \mapsto (\det A)^{-1} (a_{ij}^p) $$
is a group ...
- 6,614
6
votes
0
answers
365
views
Is this just a numerical accident or what?
In a complementary proof for a matrix determinant of $a_{i,j}=\binom{n-1+i}j$, raised by BillyJoe, I showed the more general evaluation
$$\det\left(\binom{i+p}{j+k-1}\right)_{1\leq i,j\leq m}
=\prod_{...
- 43.2k
6
votes
0
answers
122
views
Schur indices for 2-groups
I am looking for any results on Schur indices over $\mathbb{Q}$ for 2-groups. By a theorem of Roquette (corollary 10.14 in Isaacs) these numbers are at most 2. I am interested in 2-groups for which ...
- 81
6
votes
0
answers
163
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Generalisation of the Witt–Berman induction theorem
$\DeclareMathOperator{\Aut}{Aut}\DeclareMathOperator{\Ind}{Ind}$I believe I can prove the following induction theorem (modulo carefully checking a few details), and I would like to know whether this ...
- 13k
6
votes
0
answers
179
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Tensoring Harish-Chandra bimodules with Verma modules
The question is about the functor $T_\lambda$ defined by Bernstein and Gelfand in the paper Tensor Products of Finite and Infinite Dimensional
Representations of Semisimple Lie Algebras.
Setup: Let $\...
- 335
6
votes
0
answers
236
views
When is an irreducible unramified principal series representation $K$-spherical?
Let $G = \operatorname{GL}_n(\mathbb Q_p)$, $T$ the usual maximal torus of $G$, and $K = \operatorname{GL}_n(\mathbb Z_p)$.
Let $\chi$ be an unramified character of $T$, with $\chi(t_1, ... , t_n) =...
- 6,180
6
votes
0
answers
239
views
Direct sum decomposition of the space of cuspidal automorphic forms
$\newcommand{\G}{\mathbb{G}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\A}{\mathbb{A}} \newcommand{\Autom}{\mathcal{A}} \newcommand{\cen}{\mathcal{Z}} \newcommand{\lieg}{\...
- 5,382
6
votes
0
answers
1k
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Definition of Admissible Representation
Let $G$ be a connected, reductive group over a number field $k$. Let $v$ be a place of $k$.
If $v$ is finite, an admissible representation of $G(k_v)$ is defined to be an abstract representation of $...
- 6,180
6
votes
0
answers
184
views
Reference request: fusion rules for unitary dual of SL(3,R)?
By the fusion rules, I mean: given two unitary irreps of the group, what unitary irreps occur in their tensor product and with what "multiplicity"? (I am guessing that direct integrals ...
- 25.8k
6
votes
0
answers
233
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Tracking down a copy of "Mixed categories, Ext-duality and representations (results and conjectures)"
I am trying to find a copy, ideally digital, of the following preprint:
Title: Mixed categories, Ext-duality and representations
(results and conjectures)
Authors: A. Beilinson and V. Ginzburg
Year:...
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