All Questions
Tagged with reference-request rt.representation-theory
823 questions
0
votes
1
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250
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Reference Request: Irreducibles of the regular representation of the permutation group is absolutely irreducible
I am writing a paper(physics) where I am using the fact that the irreducible's of the regular representations of the permutation group are absolutely irreducible in the following sense.
If $V$ is an ...
- 349
5
votes
1
answer
436
views
Is the Veronese variety "enough" to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?
I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts.
Let $V$ be a complex vector space of dimension $n$, ...
- 2,527
5
votes
1
answer
202
views
Alternating elements in free graded-commutative algebras
It is classical that every alternating polynomial is (uniquely) the product of a symmetric polynomial with the Vandermonde polynomial, in particular the alternating polynomials are a free rank-one ...
- 17.4k
12
votes
1
answer
472
views
Growth of dimension of fixed spaces in $GL_n(\mathbb{Q}_p)$-representations
Let $\pi$ be a generic irreducible admissible representation of $GL_n(L)$, where $L$ is a $p$-adic field, $R$ is its ring of integers, and $\mathfrak{p}$ is its prime ideal. The conductor of $\pi$ ...
- 1,453
1
vote
1
answer
881
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Why is every l-adic Galois representation conjugate to one over the l-adic integers? [closed]
Why is every l-adic Galois representation
$$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Q}_{l})$$
conjugate to one over the l-adic integers?
$$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Z}_{l})$$
- 1,805
10
votes
1
answer
389
views
Operads and the Stable Module Category
I am seeking references to places where operads and their algebras have been studied for the stable module category. Colored operads are fine too.
Let $k$ be a field and $R$ a $k$-algebra. The stable ...
- 30.3k
6
votes
1
answer
476
views
Structure of Deligne-Lusztig representations $R_{T,\theta}$ for ministropic $T$ and cuspidal representations
Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$.
I'll start with a somewhat vague question and make my question more specific further down:
How do ...
- 1,453
7
votes
2
answers
405
views
The Irreducible Representations of the Sekine Quantum Groups
Here Y. Sekine introduces a one-parameter family of finite quantum groups of dimension $2n^2$. Let $n\geq 3$ be fixed and $\zeta=e^{2\pi i/n}$. Set
$$\mathcal{B}_n=\mathbb{Z}_n\times\mathbb{Z}_n=\{(i,...
- 1,027
1
vote
2
answers
867
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Is the restricted root system of a simple real Lie group irreducible?
As the title asks, is the restricted root system of a simple real Lie group irreducible?
I believe this is true but I need a reference to cite.
- 19
7
votes
1
answer
376
views
Characters of cuspidal representations
Let $\pi$ be an irreducible cuspidal representation of a semi-simple $p$-adic group $G$. It is well-known that the character of $\pi$ is concentrated in the set of compact elements in $G$.
What is ...
- 7,037
5
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0
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135
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Relative invariants of $P\oplus P^*$
Let $P$ be a $\mathrm{GL}(V)$-module, and assume that the decomposition of $P$ into irreducible submodules is known. By a relative invariant of the module $P\oplus P^*$, I mean a homogeneous nonzero ...
- 2,527
1
vote
1
answer
158
views
Graded category O for for rational Cherednik algebras, but at t=0
The paper [1] introduced the category $\mathcal{O}$ for rational Cherednik algebras $H_{t,c}(W)$. This construction is tailored for the $t=1$ case (equivalently, the $t\neq 0$ case). The general setup ...
- 1,066
6
votes
1
answer
254
views
A natural Lascoux-Schützenberger involutions on plane partitions
The Lascoux-Schützenberger involutions, $s_i$, that permute the weight of semi-standard Young tableaux are fairly known.
They satisfy some nice Coxeter relations, for example, if $v$ and $w$ are ...
- 15.8k
14
votes
4
answers
1k
views
actions of the hyperoctahedral group
I am looking for actions (i.e., permutation representations) of the hyperoctahedral group $H_n$ (also known as the group of signed permutations) studied in the literature, i.e., homomorphisms from $...
- 5,822
2
votes
1
answer
514
views
Any representation is a subrepresentation of a direct sum of the regular representation
I need a reference for the following statement:
Let $G$ be a linear algebraic group over algebraically closed field $k.$ Let $V$ be a finite dimensional $G$-module. Then $V$ is subrepresentation of $...
- 661
2
votes
0
answers
187
views
Classification of Automorphism set of a Regular graph
Let $A$ be the adjacency matrix of an $r$-regular graph $G$ with $n$ vertices (Not complete or cycle graph) . Also, let $Aut(G)$ be the set of all its automorphisms (i.e. set of permutation matrices)....
- 267
5
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0
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359
views
Examples of Rankin-Selberg L-functions from Eisenstein series
I've been digging for awhile to not much success, so I figure I would try here:
I am looking for some references which compute explicitly examples of Rankin-Selberg L-functions from the constant ...
- 3,799
4
votes
1
answer
267
views
reference request: direct product of WOT-continuous unitary representations
In an article I'm revising, I spend some time giving a self-contained proof of the following result
Let $G$ be a (Hausdorff) topological group and let $(\pi_i)$ be a family of unitary ...
- 25.8k
2
votes
0
answers
245
views
Reference request: proofs of the theorems in the paper "On the representation of the group GL(n, K) where K is a local field"
In the paper On the representation of the group $GL(n, K)$ where $K$ is a local field by Gelfand and Kazhdan, it is said that the proofs of the theorems in the paper are published in some other papers....
- 6,201
7
votes
1
answer
824
views
Infinite-dimensional admissible representations of SL(2,C)
I'm working in my research with the infinite dimensional (admissible) irreducible representations of $\mathrm{SL}(2,\mathbb{C})$ introduced by Harish-Chandra in his paper "Infinite Irreducible ...
2
votes
1
answer
150
views
Irreducible unitary representations of semidirect groups of a discrete abelian group by a discrete group
Recently in a paper we get the following result:
Let a discrete group $\Gamma$ act on a discrete abelian group $G$ by group automorphisms. Every irreducible unitary representation $\pi$ of $G\rtimes\...
- 831
35
votes
4
answers
2k
views
Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?
A $4\times 4$ symmetric matrix
$$
\left(
\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{12} & a_{22} & a_{23} & a_{24} \\
a_{13} & a_{23} & a_{33} & ...
- 2,527
9
votes
3
answers
1k
views
Catalan numbers as sums of squares of numbers in the rows of the Catalan triangle - is there a combinatorial explanation?
This question arose from an answer to my recent question How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, ... algebras?
What I need from that ...
- 18.4k
3
votes
0
answers
102
views
Localized at $p$ integral representations of finite elementary $p$-groups
Let $C_p$ be a cyclic group of prime order $p$.
Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times).
I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$.
However, ...
- 14.1k
2
votes
0
answers
156
views
Extension of the Hilbert-Mumford Criterion
Let $X$ be a smooth variety, $L$ a line bundle on $X$ and $G$ a reductive group actin on $X$ with a linearization of the action to $L$. Say we are over the complex numbers.
Both the concept of GIT ...
- 2,384
3
votes
3
answers
1k
views
A table for irreducible integral representation of finite cyclic groups
Is there such a table where the irreducible integral representations of finite cyclic groups
are listed?
Edited:
Thanks for Todd Leason's comment.Acutally,i want to know all inequivalent ...
- 593
12
votes
1
answer
1k
views
Generalization of Schur polynomials
EDIT (2018-11-05) I am slowly making a list of symmetric functions, and generalizations available online here. PDFs with overviews are available there for download.
I am making a list of ...
- 15.8k
2
votes
2
answers
535
views
Is anything known about the eigenspectrum of the regular representation of the permutation group?
I am looking for information like upper bounds on how many times any eigenvalue can occur or something like how many eigenvalues can be there in some given range. Is anything like this known?
The ...
- 1,893
1
vote
1
answer
691
views
Help finding paper: De Concini, Kac - Quantum Groups at roots of 1
I am looking for a specific paper, that I have found very difficult to trace.
C. De Concini, V. Kac - Quantum Groups at roots of 1
Specifically, the paper is cited as follows (on De Concini's ...
- 13
6
votes
3
answers
813
views
Reference request about the representations of the group $PSL_2(\mathbb{F}_q)$
Is there a review/exposition of the representation theory of $PSL_2(\mathbb{F}_q)$ ? Like an enumeration of its irreducible representations and their dimensions as a function of $q$?
- 1,893
8
votes
0
answers
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
3
votes
0
answers
362
views
Unitary representation of finite-dimensional unitary group
the question is the following. Let n,m be integers, $U(n)$ be the unitary group of $M_n(\mathbb C)$, and $\phi\colon U(n)\to U(m)$ be a continuous group homomorphism, that is moreover irreducible as a ...
8
votes
1
answer
455
views
Is there any good survey on the hook length formula and related topics?
I am recently doing some research related to the hook length formula.
The hook formula counts the number of Young tableaux of certain type.
I find there are plenty of research already been done and ...
- 393
3
votes
0
answers
313
views
References for the bicategory of ring-bimodule pairs
One of the standard examples of a bicategory is the bicategory of rings (with bimodules as 1-morphisms), which is sometimes denoted $\operatorname{Bim}$ and in other sources $\operatorname{Ring}$ (or $...
- 2,450
1
vote
0
answers
70
views
$\Gamma$ cohomology of principal series
Let $G$ be a noncompact connected real semisimple Lie group with finited center. Let $\Gamma$ be a cocompact discrete subgroup of $G$, and let $P$ be a parabolique subgroup
with Langlands ...
- 1,111
5
votes
1
answer
911
views
Why Jacobson, but not the left (right) maximals individually?
I firstly asked the following question on MathStackExchange a couple of months ago. I did not receive any answers, but a short comment. So, I decided to post it here, hoping to receive answers from ...
- 493
3
votes
3
answers
598
views
First Explicit Irreducible Representations
Although the classification of simple Lie Algebras and their representations is fully understood, I wonder whether there is some book with exhaustive tables describing explicit irreducible ...
- 2,091
2
votes
1
answer
200
views
Reference that contains examples of absolutely indecomposable representations of quivers over a finite field
Is there a reference that lists/discusses examples of absolutely indecomposable representations of quivers over a finite field (absolutely indecomposable = does not decompose into a direct sum over ...
- 2,964
16
votes
2
answers
818
views
Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$
$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power $V^...
- 38.6k
3
votes
0
answers
282
views
Galois correspondence subgroups/subsystems
In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups:
Lemma 3.16. Let $G$ be a compact group and $Rep(G)$ the category of finite ...
2
votes
1
answer
280
views
Unitary representation with fixed Casimir
Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of $G$....
- 1,111
9
votes
3
answers
435
views
How do small central extensions drop the dimension of a faithful representation?
Apologies in advance that this is a very soft question. I might be talking complete nonsense. But I hope I am talking about something that has even been studied...
I am interested in the phenomenon ...
- 2,851
7
votes
2
answers
954
views
Concise mathematical definition of the fusion product on the Verlinde ring?
The Verlinde ring of a (let us say) simply connected simple compact Lie group has as underlying additive group the Grothendieck group of representations of the central extension $\widehat{LG}$ of the ...
- 35.5k
1
vote
1
answer
291
views
Explicit deformations of pseudo representations
Let $G$ be a group (which I will be glad to consider to be the absolute Galois group of a $p$-adic field, and so satisfies Mazur's finiteness condition which appears in his paper Deforming Galois ...
- 113
2
votes
0
answers
89
views
explicit matrices for Weil ($p^2$ dimensional) representation of $Sp(4,\mathbb{F}_p)$, $p>3$
I am looking for more-or-less explicit matrices for the $p^2$ dimensional Weil representation of $Sp(4,\mathbb{F}_p)$, suitable for computer implementation. Ideally, I would like the images of the ...
- 1,639
13
votes
0
answers
523
views
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
2
votes
1
answer
180
views
Is a matrix element of a norm continuous representation always a trigonometric polynomial?
I asked a similar question for the case of compact groups not long ago in math.stackexchange. Now I understand that the answer was "yes", and I want to modify that question. This is also related to my ...
- 7,426
7
votes
1
answer
259
views
Trigonometric polynomials on non-compact and non-abelian groups
I asked this initially in math.stackexchange, but it disappeared almost immediately, so I hope it will be proper to aks this here.
Hewitt and Ross define trigonometric polynomial on a locally compact ...
- 7,426
5
votes
2
answers
332
views
Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$
This question follows up a question I asked on math.SE. This is a refinement and a reference request.
For what groups $G$ does there exist a $Z(G)$-extension of $\operatorname{Aut}G$ (call it $\tilde ...
- 2,851
13
votes
3
answers
1k
views
Characterization of Frobenius complements
I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation.
That is, a finite group $G$ is a Frobenius complement if and only ...
- 8,086