**0**

votes

**0**answers

14 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 ...

**5**

votes

**3**answers

563 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 ...

**-5**

votes

**0**answers

170 views

### Are there “adelic” L-functions? [on hold]

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 ...

**0**

votes

**0**answers

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 ...

**3**

votes

**0**answers

91 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 ...

**2**

votes

**0**answers

162 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**

votes

**0**answers

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, ...

**1**

vote

**0**answers

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* ...

**1**

vote

**0**answers

39 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**

votes

**1**answer

192 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 $G_1$ ...

**1**

vote

**0**answers

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**

votes

**0**answers

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 ...

**-1**

votes

**0**answers

50 views

### relation between characters [closed]

My assumption: $ H $ is a subgroup with index $ m $ in the finite group $ G $ & $ F $ is an algebraic closed field of characteristic zero & $ \chi $ is an irreducible $ F $-character of $ G $ ...

**-1**

votes

**0**answers

33 views

### $ \mathbb{C} $-character table of $ D_{14} $ [closed]

Is there any reference where I can find the $ \mathbb{C} $-character table of the dihedral group $ D_{14} $?

**0**

votes

**0**answers

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**

votes

**0**answers

98 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**

votes

**1**answer

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**

votes

**1**answer

81 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)} ...

**1**

vote

**0**answers

109 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**

votes

**0**answers

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**

votes

**2**answers

683 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 ...

**2**

votes

**0**answers

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 ...

**2**

votes

**0**answers

69 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 ...

**3**

votes

**0**answers

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
...

**4**

votes

**0**answers

99 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**

votes

**1**answer

361 views

### Multiplicity one theorem

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

**1**

vote

**1**answer

93 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 ...

**0**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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 ${\rm SO}(n)$ ...

**1**

vote

**0**answers

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.

**0**

votes

**0**answers

63 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 ...

**1**

vote

**0**answers

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**

votes

**2**answers

252 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 ...

**5**

votes

**1**answer

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
$$ ...

**0**

votes

**1**answer

51 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**

votes

**0**answers

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 ...

**2**

votes

**3**answers

345 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 ...

**1**

vote

**1**answer

92 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 ...

**2**

votes

**0**answers

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 $\mathfrak{k}$. ...

**0**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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 ...

**5**

votes

**0**answers

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 ...

**0**

votes

**0**answers

114 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 ...

**1**

vote

**0**answers

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 ...

**2**

votes

**2**answers

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 ...

**20**

votes

**2**answers

641 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 ...

**5**

votes

**0**answers

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. ...

**0**

votes

**1**answer

104 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 ...

**1**

vote

**0**answers

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 ...

**2**

votes

**2**answers

246 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 ...