All Questions
Tagged with reference-request rt.representation-theory
823 questions
3
votes
0
answers
75
views
While expanding Jack polynomials in monomial basis
Denote $\mathbf{z}=(z_1,\dots,z_n)$. Let $P_{\kappa}(\mathbf{z};\alpha)$ be the symmetric Jack polynomials and suppose they are expanded in terms of the monomial symmetric basis $m_{\rho}(\mathbf{z})$ ...
- 43.2k
3
votes
0
answers
91
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Are the reductions of the cuspidal characters of GL2(Fq) distinct?
Let $p$ be an odd prime and $q=p^n$ for some $n \geq 1$. If $\mathbb{F}_q$ is the unique, up to isomorphism, finite field with $q$ elements then the cuspidal representations of the group $\rm{GL}_2(\...
- 117
1
vote
1
answer
178
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Unitary representations of discrete (locally compact) groups
Let $\Gamma$ be a discrete (locally compact) subgroup of a locally compact Lie group. Let $H = L^2(\mathbb R^n, \mathbb C)$. Assume we have a complex unitary representation $$\Phi : \Gamma \to \...
- 357
2
votes
0
answers
61
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Iwahori spherical representations of GL(n) with no nonzero fixed vectors under the fixator of a panel of the affine building
Let $G$ be the group $GL(n,F)$, where $F$ is a p-adic field, and $I$ its standard Iwahori subgroup. Let $\pi$ be an irreducible smooth representation of $G$ with nonzero $I$-fixed vectors. It is well ...
- 21
4
votes
0
answers
180
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Subgroups that conjugate-cover the ambient group
Let $G$ be a finite group, and suppose that a set of proper subgroups $H_1,\dotsc,H_n$ satisfy $G=\bigcup_{g\in G}\bigcup_{i=1}^nH_i^g$, where $H_i^g$ is the conjugate of $H_i$ by $g$. In this case, ...
4
votes
0
answers
132
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Ring theoretical aspects of the DAHA
The double affine Hecke algebras (DAHA) were introduced by Cherednik in his study of Macdonald's inner product conjectures (which were solved affirmatively).
Nowdays there are many variations of the ...
- 3,318
4
votes
0
answers
70
views
Permutation matrix in terms of an $\mathfrak{su}(r)$-basis (generalised Gell-Mann matrices)
Let $V \cong \mathbb{C}^r$ be the defining representation of $\mathfrak{su}(r)$. Then the permutation on $V \otimes V$ can be expressed as
$$ P = \frac{1}{r} \, 1 \otimes 1 + \frac{1}{2} \sum_{a=1}^{r^...
- 1,996
0
votes
0
answers
54
views
Number of indecomposable modules over representation-finite hereditary algebras
Let $A$ be a finite dimensional $K$-algebra over a field $K$ that is hereditary and of finite representation type.
It is well known that they are classified by Dynkin diagrams.
For algebraically ...
- 26.5k
11
votes
1
answer
340
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Number of odd-dimensional irreducible representations of $S_n$
In this paper the structure of odd-dimensional irreducible representations of the symmetric group is described, but what is the asymptotic behaviour of the number of such representations? (Or, if it ...
- 109k
3
votes
1
answer
187
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References for Bernstein-Zelevisnky classification
I am looking for references for the Bernstein-Zelevisnky classification of irreducible representations of GL$(n,F)$ in terms of cuspidal representations. In particular I would like to find something ...
- 367
5
votes
2
answers
218
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Decomposition of symmetric powers of the fundamental representation of $\text{Sp}(2n,\mathbb{C})$
Let $(k,0,...,0)$ denote the highest weights vector of an irreducible representation of $\text{Sp}(2n,\mathbb{C})$. I read in Fulton-Harris, that this representation may be obtained as a direct ...
- 2,907
5
votes
0
answers
147
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Tensor product of a Verma module of the highest weight and a Verma module of the lowest weight, $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$
$\DeclareMathOperator\sl{\mathfrak{sl}}\newcommand\hw{\mathrm{hw}}\newcommand\lw{\mathrm{lw}}$Consider $\mathfrak{g}=\sl_2(\mathbb{C})$. Fix $\lambda,\mu\in\mathbb{C}\setminus \frac{1}{2}\mathbb{Z}_{\...
- 51
1
vote
1
answer
70
views
Classifying lisse conformal vertex algebras using singularities of associated varieties
For the sake of keeping terminologies consistent, let me say that a conformal vertex algebra is a vertex algebra (VA) with a specified conformal vector, and a vertex operator algebra (VOA) is a ...
- 1,516
2
votes
0
answers
97
views
Centralizer bound for irreducible representations of $\operatorname{SU}_n(\mathbb{C})$
Let $G$ be a finite group, $χ$ be the character of an irreducible representation $V$ of $G$, and $g ∈ G$. Then a classical bound on the trace of $g$ is given by: $|χ(g)|² ≤ |C_G(g)|$, where $C_G(g)$ ...
- 2,907
7
votes
0
answers
141
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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
16
votes
0
answers
188
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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
2
votes
1
answer
78
views
Reference for irreducible representations of $\mathcal{O}(n)\ni O\mapsto O^{\otimes k}$
This MO answer cites the Goodman-Wallach book to affirm that:
$$\mathrm{Sym}^k\left(\mathbb{R}^n\right)=\mathcal{H}^k\oplus q\mathcal{H}^{k-2}\oplus q^2\mathcal{H}^{k-4}\oplus\cdots$$
with $\mathrm{...
- 167
1
vote
0
answers
172
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Isomorphism classes of finite $\mathbb{N}$-groups
Where can I find resources on isomorphism classes of finite $\mathbb{N}$-groups, i.e. groups acted on by the monoid $(\mathbb{N}, +)$?
I edited this question to be more focused on what I'm interested ...
- 591
2
votes
1
answer
177
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Action of $O(3,\mathbb{R})$ on the conic $\{x^2+y^2+z^2=0\}$
The action of the orthogonal group $O(3,\mathbb{R})$ on the conic
$C= \{ x^2+y^2+z^2=0 \}$ in $\mathbb{P}^2$ must be well-understood, but I could not find any reference.
Is it doubly transitive?
- 14k
11
votes
1
answer
383
views
Is there a comprehensive survey of the discrete series representation of a real reductive group?
Vague form of the question: where can one find a comprehensive and possible modern account of the discrete series representations of a real reductive group?
Motivation:
I am a master's student trying ...
- 511
2
votes
1
answer
131
views
Extending $p$-adic smooth and locally constant functions
Let $G$ be $p$-adic group and let $G \rightarrow GL(V)$ be a representation. For example, $V$ is a quadratic $\mathbb{Q}_p$-space and $G$ is the associated orthogonal group.
Take a point $v \in V$, ...
- 81
1
vote
0
answers
98
views
Are there known effective bounds on the number of semisimple Galois representations?
In continuation to my question here, are there known effective bounds on the total number of semisimple $p$-adic Galois representations unramified outside a finite set of primes $S$, of dimension $d$, ...
- 2,907
1
vote
1
answer
114
views
Source for highest weight vectors for $\text{SL}_n(\mathbb{C})$ representations
I have a list of a lot of irreducible $\text{SL}_n(\mathbb{C})$ representations and need to know what all of their highest weight vectors are. For example, using the notation from Fulton and Harris, I ...
- 181
5
votes
0
answers
140
views
Classification of visible actions for *reducible* representations?
Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...
- 3,230
12
votes
2
answers
836
views
Restriction of $\mathrm{GL}(n)$ representation to $S_n$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\O{O}$I'm looking for a reference to cite for the following observation. Given an irreducible representation of $\GL(n)$ labelled by the Young diagram ...
- 21.5k
2
votes
0
answers
118
views
What are the finite-dimensional irreducible unitary representations of $E(3)$?
Let $E(3)$ be the Euclidean group of $\mathbb{R}^3$ defined, e.g., by
$$E(3)=SO(3)\ltimes T(3)$$
where $T(3)$ is the translation group.
I am looking for a reference classifying all the finite-...
- 31
2
votes
0
answers
49
views
Deformed preprojective algebras of generalized Dynkin type
Question 1:Is it true that the selfinjective (finite dimensional over an algebraically closed field K) algebras $A$ such that the stable module category of $A$ is 2-Calabi-Yau are exactly the deformed ...
- 26.5k
6
votes
1
answer
274
views
Subrepresentations of the $\text{SL}_n(k)$-representation $\mathfrak{gl}_n(k)$
$\DeclareMathOperator\SL{SL} \newcommand{\gl}{\mathfrak{gl}} \newcommand{\sl}{\mathfrak{sl}}$One of my graduate students asked me for a reference for the following fact. Let $k$ be a general field (...
- 44.8k
1
vote
0
answers
172
views
Representation theory of $\mathrm{GL_n}(\mathbb{F}_q)$
I am interested in learning about the classification of irreducible representations of $\mathrm{GL}_n(\mathbb{F}_q)$ as done by Green. One standard reference (besides Green's original work) is ...
- 697
5
votes
1
answer
187
views
Reference Request: Test vectors for local Rankin-Selberg L-factors in ramified cases
Let $F$ be a global number field, i.e. a finite extension of the field of rational numbers. Let $\sigma$, $\pi$ be automorphic representations of $\mathrm{GL}_n(F)$ and $\mathrm{GL}_{n+1}(F)$ ...
- 639
0
votes
0
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117
views
An interesting identity involving skew-Schur functions
Denote $\rho=(-\frac12,-\frac32,-\frac52,\dots)$. I was reading this interesting paper, where in particular, the authors claim that one can get the expression in (2.9)
\begin{align*}
\prod_{k\geq1}(1+...
- 43.2k
16
votes
3
answers
1k
views
Conjectures in the representation theory of the symmetric group
Question: What are current open conjectures about the representation theory of the symmetric group?
I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
- 26.5k
1
vote
0
answers
47
views
Absolute irreducibility implies free action on framed universal deformation ring
Let $\overline{\rho}: G\longrightarrow \text{GL}_n(\mathbb{F}_p)$ be a residual representation of the Galois group of a number field. Let $R_{\overline{\rho}}^{\square,\text{univ}}$ be the universal ...
- 2,907
3
votes
0
answers
107
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Representations of a reductive Lie group vie central character and K-types
Let $G$ be a real reductive group, let $\widehat G$ denote the unitary dual and $\widehat G_{adm}\supset\widehat G$ be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-...
- 460
1
vote
0
answers
124
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Computing the induced homomorphisms of derived functors using acyclic resolutions
Let's suppose that $F\colon \mathcal A\to \mathcal B$ is a right exact additive functor between abelian categories such that $\mathcal A$ has enough projectives. Standard references shows that if $Q_\...
- 38.6k
1
vote
0
answers
158
views
What is the "weight" of an automorphic form for $\mathrm{PGL}_2$?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I'm trying to understand what the notion of "weight" is for automorphic forms over $\GL_2(F)$ where $F$ is some number field, in ...
- 423
1
vote
0
answers
148
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Exploring the Intersection of Expander Graphs, Number Theory, Representation Theory and Recent Computer Science Developments [closed]
I have a solid understanding of the basics of expander graphs and their properties and the recent development of High-Dimensional Expanders and their application to Random Walks, along with other ...
1
vote
1
answer
155
views
Iwahori action on the $p$-ordinary line of a principal series representation
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}\newcommand\Iw{\mathrm{Iw}}\DeclareMathOperator\ord{ord}$Let $F$ be a $p$-adic local field, i.e. a finite ...
- 639
3
votes
1
answer
238
views
Steenrod operations on classifying spaces
Steenrod operations can be defined for all finite characteristics $p$. The simplest one, when $p=2$, is the Steenrod square. I wonder if the computation for classifying spaces for classical Lie groups ...
- 803
1
vote
0
answers
127
views
Irreducible projective representations of finite abelian groups
I want to know if there is a description of all irreducible complex projective representations of an arbitrary finite abelian group. I have seen this for particular cases such as those given here and ...
- 249
4
votes
0
answers
183
views
Characters of finite fields - Reference request
Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of ...
- 41
4
votes
0
answers
168
views
Representations of $C\left(SO_q(n)\right)$
A complete classification of irreducible representations of the $C^*$-algebra $C(G_q)$, where $G_q$ is the $q$-deformation of a classical simply connected semisimple compact Lie group, was provided by ...
- 73
1
vote
0
answers
110
views
Character table of $\mathrm{P\Gamma L}_2(q)$ with $q$ even
Let $q = 2^f$ for some integer $f\geqslant 3$. The character table of $\mathrm{SL}_2(q)\cong\mathrm{PSL}_2(q)$ can be deduced from the character table of $\mathrm{GL}_2(q)$ (see, for example, Exercise ...
- 379
11
votes
1
answer
482
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What is the commutative coproduct and where can I learn more about it?
This is a reformulation of a Mathematics Stack Exchange question that got no answer there, and, at the same time, a follow-up to another question on MSE.
The original problem was to prove $U(\mathfrak{...
- 491
1
vote
1
answer
232
views
Transfer for the group of coinvariants: a reference request
Let $G$ be a group and $M$ be a $G$-module,
that is, an abelian group written additively on which $G$ acts:
$$ (g,m)\mapsto g m.$$
We consider the group of coinvariants
$$ M_G:=G/\langle g m -m\ |\ g\...
- 14.2k
9
votes
0
answers
254
views
An identity for characters of the symmetric group
I am looking for a reference for the identity
$$\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
5
votes
0
answers
146
views
On Soergel's results concerning projectives modules in category $\mathcal{O}$
I am looking for a translated proof of two results of Soergel usually referred to as endomorphismensatz and struktursatz.
Both of those results were shown in the paper
Soergel, W. (1990). Kategorie 𝒪...
- 211
2
votes
0
answers
352
views
On characters of the symmetric group: Part 1
Given an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\lambda\vdash n$ either as $(\lambda_1,\dots,\...
- 43.2k
1
vote
0
answers
126
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Reference request: unfolding of Integral representation of an L-function
Are there any text or papers that thoroughly address unfolding of integral representation of an L-function such as D. Ginzburg's On Spin L-function for Orthogonal Groups page 762-763 and page 774 (or ...
- 43
3
votes
1
answer
327
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Pro-unipotent radical (in Bruhat-Tits) vs. unipotent radical (and reference request)
I'm a beginner in Bruhat-Tits theory, and the following phenomenon makes me puzzled.
So let $F$ be a $p$-adic field, with $\mathfrak{o}\supset \mathfrak{p}$ its ring of integers and maximal ideal, and ...
- 709