Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

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121 views

Algebraic groups “generated” by a Lie algebra element

Here is a definition which I invented and which I would like to understand better. Let $ A $ be a complex affine algebraic group. Let $ X \in \mathfrak g $ be an element in its Lie algebra. We say ...
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33 views

Pseudo-braided fusion categories

A few definitions first, please replace with the standard terminology (and correct me if I confuse all the by-names of fusion categories :-) I call a complex number $z$ pseudo-cyclotomic if $|z|=1$. I ...
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0answers
79 views

Two questions about Whittaker functions

I am watching the video: Modeling p-adic Whittaker functions, Part I. I have two questions about Whittaker functions in the video. From 33:00 to 37:00, it is said that after changing of variables, ...
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0answers
19 views

Tensor Algebra Quotients of the Ext-Alg of an SU(2)-Module

Let $V$ be a simple (left) $SU(2)$-module, and $T(V)$ the tensor algebra of $V$. If we quotient $T(V)$ by the ideal generated by a simple submodule of $V \otimes V$, is there a general system for ...
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3answers
643 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 ...
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173 views

Are there “adelic” L-functions? [closed]

Following Tom163's answer to this question, I would like to know whether L-functions defined through adelic representations (as defined in https://projecteuclid.org/euclid.em/1317758108) have been ...
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40 views

Non-trivial summand in End(\rho)

Given a finite group representation $\rho:G\to GL_n(\mathbb C)$ one knows that the trivial representation $\mathbb 1$ is contained in $End(\rho)$. Let $\rho'$ be the other summand, i.e., $\rho'$ is ...
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94 views

n-homology of a Harish-Chandra module

Let $G$ be a connected real reductive Lie group and let $K$ be its maximal compact subgroup. Let $P=MAN$ a parabolic subgroup. Let $K_M^0=M^0\cap K$ be connected component of the maximal compact ...
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163 views

References for 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'

Presently I am reading the 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'. I am finding it difficult for eg. the initial sections on $l$-adic geometric representation of finite ...
3
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78 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, ...
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27 views

Fusion category R-symbols diagonal even when multiplicity present?

Preface: Please bear my lousy ASCII :-) I automatically thought that a twist move on a trivalent node -(/ = a* -( would be diagonal even in the case of multiplicity, for example: -1(/= a* -1( + 0* ...
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40 views

Character sums over a fixed subset of skew tableaux

Let $f(\lambda)$ count the number standard young tableaux of shape $\lambda\vdash n$ and $\lambda=(\lambda_1,\cdots,\lambda_r)$. Let $\mu \vdash k$ be a partition for $k<n$. It is a consequence of ...
5
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1answer
211 views

real representation of a product group

Let $G_1$ and $G_2$ be compact Lie groups. We know that each finite-dimensional complex irreducible representation of $G_1\times G_2$ is the tensor product of an irreducible representation of $G_1$ ...
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0answers
54 views

Symmetric and antisymmetric powers of SU(2) representations [closed]

Recently, I took a course in representation theory at Imperial College, and on the first homework the questions were about certain sneaky relationships when it came to representations of SU(2). ...
2
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0answers
87 views

Generating free groups by small subgroups and an element

Let $F$ be a free group of countably infinite rank, and let $L \leq F$ be a finite index subgroup. For some prime $p$ and $k \in \mathbb{N}$, let $\sigma \colon L \to ...
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117 views

Reference request: the formula $\langle x, [f, g] \rangle = \langle \delta(x), f \otimes g \rangle$

Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}^*$ the dual vector space of $\mathfrak{g}$ which is also a Lie algebra with natural brackets. Let $\delta: U(\mathfrak{g}) \to U(\mathfrak{g}) ...
3
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100 views

Nilpotent orbits and subspaces

Let ${\mathbb g}$ be a simple complex finite dimensional Lie algebra, $X\subseteq{\mathbb g}$ a nilpotent orbit. Did anyone study maximal vector subspaces of the closure $\overline{X}$? In ...
5
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1answer
91 views

For what $G$ is $Rep(D(S_3))_{ad}$ Grothendieck equivalent to $Rep(G)$?

Given a fusion category $\mathcal C$, the Grothendieck Ring $K_0(\mathcal C)$ is the $\mathbb Z$-based ring whose basis elements correspond to isomorphism classes of simple objects and whose ...
5
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1answer
82 views

Sum identities with immanants

For $\chi$ being an irreducible character of the symmetric group $S_n$ and being $M$ a complex $n\times n$-matrix, I would like to show $$ \sum_{\sigma, \rho \in S_n} \overline{\chi(\sigma)} ...
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0answers
112 views

Representations and K-theory of a finite group

This question is motivated by the calculation of the higher algebraic $K$-groups of a finite field. Let $G$ be a finite group, the case I am most interested in is $G = \text{Gl}_n(\mathbb F_q)$, but ...
3
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0answers
35 views

Characterizations of Jacobson-Morozov parabolics associated to a nilpotent

Let $x \in \mathfrak{g}$ (or $x \in G$) be a nilpotent (resp. unipotent) element of a simple Lie algebra (resp. linear algebraic group). One can associate to this data a Jacobson-Morozov parabolic ...
16
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2answers
687 views

How bad can $\pi_1$ of a linear group orbit be?

Let $G$ be a simply connected Lie group and $\mathcal O= G(v)=G/G_v$ a $G$-orbit in some finite-dimensional $G$-module $V$. By the homotopy exact sequence, its fundamental group $\Gamma$ is the ...
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0answers
77 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 ...
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70 views

Are Markov traces matrix traces?

When starting this question I was very hesitant - literature on the subject is vast and I thought most likely the answer is already there somewhere. Then when the list "Questions that may already ...
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0answers
92 views

Ext and representations with fixed central characters

In this paper (http://arxiv.org/pdf/1108.3668v2.pdf) Adler and Prasad compute certain Ext groups. On page 2 they write, "Since extensions of representations of abelian groups are well understood ...
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100 views

Young Tableau Box Correlations

Let $T$ be a uniformly random Standard Young Tableau (SYT) of shape $\lambda=(\lambda_1,\cdots,\lambda_k)$ with $|\lambda|=n$. Let $T_{ij}$ denote the value in box $(i,j)$. I'm interested in what can ...
6
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1answer
362 views

Multiplicity one theorem

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) ...
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1answer
95 views

Finite groups normalizing a torus

Let $G$ be a semi-simple linear algebraic group over the complex numbers, e.g. the special linear group. Can you find an example of a finite sub-group $H$ of $G$ which does not normalize any maximal ...
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41 views

Semisimple Coquasitriangular Hopf Algebras

Let $(G,R)$ be a coquasitriangular Hopf algebra, which we furthermore assume to be cosemisimple. Generalising classical Peter-Weyl, cosemisimplicity of $G$ is well-known to imply an isomorphism $$ G ...
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0answers
101 views

What are the general zonal spherical functions for ${\rm SO}(n)/{\rm SO}(n-1)$?

The zonal spherical functions [1] on the sphere $(G={\rm SO}(n)$, $K={\rm SO}(n-1))$ are the Gegenbauer or ultraspherical polynomials if one considers the irreducible representations of ${\rm SO}(n)$ ...
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85 views

What are the differences between p-adic Whittaker functions and archimedean Whittaker functions? [closed]

What are the differences between p-adic Whittaker functions and archimedean Whittaker functions? Are there some references about the differences? Thank you very much.
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64 views

What are the module categories with finitely many modules with trivial endomorphism ring?

If an finite-dimensional k-algebra is representation finite, there are also finitely many endotrivial (finite-dimensional) modules M, i.e. with trivial endomorphism ring Hom(M,M)=k.There are ...
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60 views

Representations of Hamilton's real/complex quaternions algebra

A lot of works and questions deal with classifying representations of a simple central algebra of given dimension over a non-archimedean field, for instance here. But do we know precisely such a ...
6
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2answers
254 views

Strategies for proving a category is Noetherian?

Let $C$ be a small linear category over a commutative ring $R$. A representation of $C$ is an $R$-linear functor $C \to \mathrm{Mod}(R)$. For example, for each $c\in C$, there is a representation ...
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1answer
210 views

Dimension of the span of all partial derivatives of a given symmetric polynomial $f$ and the polynomial $E(f)$

I need some help on the problem below. Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set $$ ...
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1answer
52 views

Non Lie-group ribbon categories

I learnt here that a) Reshitikhine-Turaev works with any ribbon category but b) those not coming from Lie groups are rare. Can someone give an actual example (and preferrable with purely graphic ...
3
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0answers
91 views

scalar multiple of Young symmetrizer

The following is a lemma from Fulton and Harris' book -Representation theory,a first course (page 53): Lemma: For all $x\in \mathbb{C}\mathfrak{S}_r$, $c_{\lambda}\cdot x\cdot c_{\lambda}= scalar ...
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3answers
346 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 ...
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1answer
93 views

Projectors onto the invariant subspaces of a unitary representation $U \otimes U^* \otimes U \otimes U^*$

Let $$U \mapsto U \otimes U^* \otimes U \otimes U^*$$ be a unitary representation of the unitary group $U(n)$ acting on the vector space $V$ (where $U^*$ is the complex conjugate of $U$). We can ...
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0answers
73 views

Contraction of the maximal submodule in a Verma module

Suppose $\mathfrak{g}$ is a real semisimple Lie algebra, $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ is a Cartan decomposition, and $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{k}$. ...
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46 views

the complex representations of $B(2, \overline{\mathbb{F}_p})$

as the title, I want to know the complex representations of the $B(2,\overline{\mathbb{F}_p})$, i.e. invertible upper triangle matrix groups over $\mathbb{F}_p$'s algebraic closure ...
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89 views

Are such averages known with representations of $S_n$?

Like is there a sense in which one can quantify that for two group elements (in different conjugacy classes) their characters are "close" for some fixed irreducible representation? (feel free to ...
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71 views

What is the importance of the number $k+h^{∨}$ (level+dual Coxeter number)?

The number $k+h^{∨}$ appears at many places in the representation theory of affine Lie algebras (and probably elsewhere). Here $h^{∨}$ is the dual Coxeter number of the root system, and $k$ is the ...
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115 views

Symmetric kernel of tensor product

Let $V,W$ be two vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with disjoint kernels $K_i$ of dimension $1$. Consider the tensor product of these maps $L_1\otimes ...
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80 views

Generalization of Schur polynomials

I am making a list of generalizations of Schur polynomials and other closely related polynomials that appear in representation theory. My motivation is to eventually make a nice poster of ...
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2answers
89 views

How does one calculate/estimate/guarantee the girth of a non-Abelian Cayley graph?

This question is in reference to this other question, Can someone point out references (or explain!) which give techniques of being able to prove for any Cayley graph this property of having a girth ...
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2answers
644 views

Intuition behind the definition of quantum groups

Being far from the field of quantum groups, I have nevertheless made in the past several (unsuccessful) attempts to understand their definition and basic properties. The goal of this post is to try to ...
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218 views

Koopman representation, weakly compact action, Ozawa Popa

Given a weakly compact action (Ozawa-Popa) of a discrete group $\Gamma$ on p.m space $X$, consider the Koopman representation $\pi$ on $L^2(X)$. Compose this representation with the Calkin projection. ...
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1answer
105 views

Irreducible representation of $C^*(D_\infty)$, group $C^*$-algebra of an infinite dihedral group

I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$. Ultimately, I'm interested in finding a ...
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110 views

How generic are Cayley graphs of non-Abelian groups with logarithmic girth?

Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know, For which $G$ can the ...