All Questions
Tagged with reference-request rt.representation-theory
823 questions
3
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Reference for Clifford theory of algebras
Clifford theory relates the representation theory of a group to that of a normal subgroup. A good reference for this is Curtis and Reiner's "Methods in Representation theory II", Theorem 11.1.
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- 1,413
6
votes
1
answer
828
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Which functions are linear combinations of irreducible characters for a given field $\Bbbk$?
Let $G$ be a finite group. Then it is well known that a function $f\colon G\to \mathbb C$ is a linear combination of irreducible characters iff it is constant on conjugacy classes. What is the ...
- 38.6k
1
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1
answer
353
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Decomposition of an induced representation
If there is a finite group $G$ with a cyclic normal subgroup $C_n$, one can describe the indecomposable representations of $G$ through induction. How does $Ind_{C_n}^G$ decompose? For representations ...
- 13
6
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3
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1k
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Good book on representation theory of GL(n)
I am interested in a recommendation for a good book which discuses representation theory of GL(n)(say over field of complex numbers).
I know only a basic representation theory.
The question I am ...
4
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3
answers
2k
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Lie subgroups of SU(4)
Other than subgroups of SU(3), what are the Lie subgroups of SU(4)? Assume that the subgroup is closed but not necessarily connected.
Additionally, which of these subgroups admit four dimensional ...
- 141
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2
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1k
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Origin of notion of "split Grothendieck group"?
In the construction of Soergel's bimodules in representtion theory , it's essential for him to work with split Grothendieck groups. Here he starts with a certain small additive category $\mathcal{A}$...
- 52.9k
1
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2
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557
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Is there formula name and proof for this theorem ? [closed]
The formula answers: how many tuples $(\sigma_1,\sigma_2,...,\sigma_n)$ of elements of a given group G such that
(1) $\sigma_i\in C_i$ , where $C_i$ stands for conjugacy class.
(2) $\sigma_1\...
- 11
7
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1
answer
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What is the support of the Whittaker function of a new vector on GL(2)?
Let $W$ be the normalized Whittaker function associated to a new vector in an irreducible generic representation $\pi$ of $G=GL_2(k)$, where $k$ is a $p$-adic field. Let $c$ be the conductor of $\pi$, ...
- 3,183
7
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2
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595
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Representation theory of Discrete Subgroups of Lie groups
My question is the following. Which representations of $Sp(2g, \mathbb Z)$ are extendable to representations of $Sp(2g, \mathbb C)$ or $Sp(2g, \mathbb R)$. Is there a general theory and a good ...
- 327
5
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3
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578
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Non-symmetric quiver varieties
Given a symmetric Cartan datum $(I,\cdot)$, H. Nakajima has defined a family of varieties - known as quiver varieties - and has used them to give geometric constructions of the representation theory ...
- 1,201
6
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1
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434
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Previous work on this generalization of continued fractions?
The 2x2 matrix representation of a continued fraction makes it clear that we're multiplying together a bunch of group elements. Inversion is essentially freely adjoining a generator to a Coxeter ...
- 1,532
3
votes
1
answer
1k
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Finite subgroups of SO(3)
There are several proofs of the famous classification of finite subgroups of $SO(3)$. I heard that there is a "purely algebraic" one attributed to Camille Jordan. Does anybody know of a reference?
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- 607
3
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0
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264
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How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?
This is a crosspost from MSE since I haven't found an answer there yet.
I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use $\mathbb{F}...
- 1,173
19
votes
2
answers
2k
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Dual versions of "folding" symmetric ADE Dynkin diagrams?
Start with the Dynkin diagram of an irreducible root system, typically associated with a simple
Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE
diagrams ...
- 52.9k
4
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1
answer
686
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Character theory of $2$-Frobenius groups.
This is a crosspost of my (slightly longer) question on MSE since I'm not getting any responses there.
Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\left(G/...
- 1,173
5
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1
answer
695
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Convex PBW bases
Given a reduced expression for the longest word $w_0$ in the Weyl group of $\mathfrak{g}=\mathfrak{n}^+\oplus\mathfrak{h}\oplus{n}^-$, one obtains a convex ordering on the set of positive roots, $\...
- 1,472
33
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2
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2k
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What do cluster algebras tell us about Grassmannians?
One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
- 1,690
12
votes
1
answer
1k
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Smallest dimension of nontrivial representation of a simple Lie algebra over `$\mathbb{C}$`
The question involved here is natural and very classical, but I'm unsure what has been formally stated and proved in the literature. The only approach I know involves assembling facts that apparently ...
- 52.9k
31
votes
1
answer
5k
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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
5
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3
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330
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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 $\...
- 453
5
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2
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923
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Status of a conjectural definition of H. Nakajima
In his paper '$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 $...
- 1,201
10
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1
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648
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The Fukaya category of a simple singularity (reference request)
I have heard that for an ADE singularity $f$,
$ D^b\mathrm{Fuk}(f) \simeq D^b(\mathrm{Rep}\ Q)$
where $Q$ is the corresponding Dynkin quiver. (As one would hope, if $\mathrm{Fuk}$ is some kind of ...
- 853
5
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1
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421
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Rational automorphisms of semisimple algebraic groups
Suppose $G$ is a semisimple algebraic group defined over a field $k$. Let $\mathrm{Aut}(G)$ and $\mathrm{Inn}(G)$ denote the groups of automorphisms and inner automorphisms (respectively) of $G$. ...
- 583
4
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0
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386
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Reference request for character formula between tensor products of Weyl modules.
So it is well known that when you tensor together two induced modules for an algebraic group
$\nabla(\lambda) \otimes \nabla(\mu)$ that the result has a filtration by other induced modules, (I.e. it ...
- 41
8
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1
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468
views
Explicit method to compute Macdonald/Koornwinder functions
I'd like to compute explicitly symmetric Macdonald functions associated to arbitrary (possibly non-reduced) root systems, using Computer Algebra System.
Unfortunately Sage seems to only implement ...
- 6,123
11
votes
1
answer
397
views
Is there a Dedekind-Frobenius group determinant for infinite groups?
If $G$ is a finite group and $\lbrace x_{g} \rbrace_{g\in G}$ are commuting formal variables, then one can form a matrix whose $(g,h)$ entry is $x_{gh^{-1}}$. The determinant of this matrix is a ...
- 7,067
4
votes
2
answers
505
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comprehensive presentation of the unitary dual of $SO_0(n,1)$
The unitary dual (unitary irreducible represenations) is determined for every connected noncompact semisimple Lie group of real rank one. I would like to have a reference for the particular case $SO_0(...
- 2,446
7
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1
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1k
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Are irreducible representations subrepresentations of a symmetric power representation?
First of all I am far from being an expert in representation theory, so it is possible (likely) that the following question is trivial (in fact a trivial reference question):
Let $\Gamma$ be a, let's ...
- 1,939
16
votes
2
answers
992
views
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 $G$ ...
- 2,179
26
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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 ...
- 52.9k
4
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0
answers
323
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The proof of the splitting principle in equivariant K-theory via flag manifolds
In Atiyah's famous paper "Bott periodicity and the index of elliptic operators" section 4, he proved the splitting principle for unitary groups (Propostion 4.9 in that paper), namely:
Let $j: T\...
- 9,019
1
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1
answer
666
views
Conjugacy classes in Aut(G)
Let $G$ be a connected, simply-connected simple group over $\mathbb{C}$. The structure/classification of conjugacy classes of $G$ is described in many places.
Now, I'd like to know the structure/...
- 6,123
6
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0
answers
168
views
Classifying algebras with two idempotent generators and involution
Say that a finite-dimensional algebra $H$ over a field $K$ is dihedral if $H$ is generated by idempotents $P_1$ and $P_2$ and there is an algebra involution interchanging $P_1$ and $P_2$.
For example,...
- 1,021
6
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1
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2k
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A request for suggestions of advanced topics in representation theory
Please Note: The main points of the question below are in bold in order to minimize the time required to read the question.
Let me begin by stating that I understand representation theory is a vast ...
15
votes
1
answer
954
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Funktorialität in der Theorie der automorphen Formen
In 2010 Langlands wrote an article with the title Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele. On the IAS website, he says that
This note ... was written ...
- 18.7k
12
votes
2
answers
541
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S-matrix for the HOMFLY/Hecke category
This question concerns the HOMFLY-PT category, closely related to Hecke algebras. (See here for example.)
The minimal idempotents of this category are indexed by pairs $(\lambda_+, \lambda_-)$ of ...
- 12.8k
6
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0
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202
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S-matrix for the BMW category
This question concerns the Birman-Murakami-Wenzl category, or equivalently the tangle category associated to the 2-variable Kauffman polynomial. (See here for example.)
The minimal idempotents of ...
- 12.8k
2
votes
4
answers
886
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A reference book for Schur's lemma
We want to find the reference book for some version of the Schur's lemma which covers the following result
Let $A$ be an assoiative algebra over $\mathbb{C}$ with countable basis, then any central ...
- 75
4
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5
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2k
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searching for text for studying representation theory
I'm a graduate student studying algebraic geometry.
Recently, When I studying Hodge theory, I saw sl2-representation is used in Hodge theory.
So I think that studying representation theory may be ...
2
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1
answer
213
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Positive definite functions on G from Hilbert space vectors?
Let $G$ be a countable discrete group. Given a vector $\xi \in l^{2}(G)$, is there any way to naturally construct a positive definite function on $G$ using $\xi$?
This question is rather vague and ...
2
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1
answer
736
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Schur Weyl duality for sl_n representations
Consider a finite dimensional vector space $V$ and the general linear group $GL(V)$ acting on it. Both $GL(V)$ and the symmetric group $S_d$ act on the tensor product of $d$ copies of $V$, and by Weyl ...
- 596
11
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4
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2k
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Textbook source for finite group properties deducible from character table?
Various questions have been posted on MO (some answered, some not) involving the character table of a finite group $G$ over a splitting field such as $\mathbb{C}$ of characteristic 0. My basic ...
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5
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0
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Is there a brief name for the symmetric space $SL_{2n} / Sp_{2n}$?
Let $V$ be a complex vector space of even dimension. Then the homogeneous space $SL_{2n}(V) / SP_{2n}(V)$ is known to parametrize the space of non-degenerate skew-symmetric bilinear forms on $V$.
(1)...
10
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3
answers
1k
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Subgroups of GL_2 over a finite field
I've come across the phrase "by the classification of subgroups of $GL_2(F_q)$" in multiple papers, but never with a reference. Here $F_q$ is a finite field of size $q$. Does anyone know a good ...
- 101
2
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1
answer
632
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Pseudo coefficients and orbital integrals
I am looking for a reference/idea, how this passage from Labesse's Snowbird Lecture "Introduction to endoscopy" pg.5 can be explained:
"We shall denote by $f_\pi$ a pseudo-coefficient for $\pi$, ...
- 11.2k
8
votes
0
answers
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
6
votes
2
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856
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Algorithm for Brauer lifting via Brauer tree?
Background: Given a finite group $G$ and a prime $p$ dividing its order, Brauer theory compares the ordinary characters of $G$ with the Brauer characters arising from $p$-modular representations. On ...
- 52.9k
10
votes
2
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655
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Has there been any application of tensor species?
Joyal's combinatorial species, endofunctors in the category of finite sets with bijections $\mathbf B$ have found numerous applications. One generalisation is given by so-called "tensor species" (...
- 5,822
4
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1
answer
565
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Connection between Deligne-Mostow monodromy and Gassner representation at roots of unity of the pure braid group
I am looking for a specific reference to the connection between [1] the Deligne-Mostow monodromy and [2] Gassner representation at roots of unity of the pure braid group. I have seen many references ...
- 11.2k
2
votes
0
answers
109
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Reference for a dual version of the Cauchy decomposition.
By "Cauchy decomposition" I mean the following identity, both sides in which are representations of $GL_n(\mathbb C)\times GL_m(\mathbb C)$:
$$\mathrm{Sym}^p(V\otimes W)=\bigoplus_{\lambda\vdash p} V^\...
- 3,513