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

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6
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1answer
273 views

Are the distributive permutation groups linearly primitive?

An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$. It is called primitive if it is transitive and preserves no ...
8
votes
0answers
118 views

Intersection of Springer fibre and Schubert cell

Let us consider intersections of Springer fibres and Schubert cells in type A. Let $ Y : \mathbb C^n \rightarrow \mathbb C^n $ be a nilpotent operator. Let $$ F_Y = \{ V_0 = 0 \subset V_1 \subset ...
0
votes
1answer
127 views

The coproducts $\mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ and $\mathbb{C}[U] \to \mathbb{C}[U] \otimes \mathbb{C}[U]$

A coproduct $\varphi: \mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ is given by: $x \mapsto 1 \times x + x \otimes 1$, where $x$ is a generator of $\mathbb{C}_q[U]$. There is a ...
0
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0answers
37 views

References about reality of minimal affinizations of quantum affine algebras

Let $U_q(\widehat{\mathfrak{g}})$ be the quantum affine algebra associated to a complex simple Lie algebra $\mathfrak{g}$. A simple module $M$ of $U_q(\widehat{\mathfrak{g}})$ is called real if $M ...
4
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0answers
98 views

Papers/Programs for computing periodic KL polynomials?

Periodic Kazhdan-Lusztig polynomials (for an affine Weyl group) are polynomials that control Jordan-Holder multiplicities for certain representations ("baby Verma modules") of an algebraic group in ...
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99 views

Connection between Lie algebras and fusion rings

Example: Take the irreps of $SU(2)$: $0,1/2,1,...$ (Spin notation.) The quantum dimensions are $1,q+1/q,q^2+q+1+1/q+1/q^2,...$. At $q=(-1)^{1/5}$ this evaluates to $1,\phi,0,...$ and you get the ...
5
votes
1answer
168 views

What is the Schur multiplier of the affine linear group AGL(n,q)?

What is the Schur multiplier of the $n$-dimensional affine linear group $\mathrm{AGL}(n,q)$ over the Galois field with $q$ elements? I am particularly interested in the simple case $n=1$. Computation ...
8
votes
1answer
216 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 ...
4
votes
1answer
147 views

A small rank linear combination of a small number of elements of a group

This is a version of this question of Klim Efremenko. Let $r>2$ be a natural number, say $r=3$ or $r=10$. Let $G$ be a finite group and $\rho$ be an irreducible complex representation of $G$. We ...
4
votes
2answers
113 views

Measurable representations of semi simple Lie groups

Let $G$ be a semi simple Lie group. I'm particularly interested in $SL(n,\mathbb{R})$. It is proved in I. E. Segal and J. von Neumann, A theorem on unitary representations of semisimple Lie groups, ...
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0answers
36 views

Approximating eps-homomorphisms

Let $G$ be a finite group. A map $\rho:G\rightarrow U_n$ is called an $\epsilon$-homomorphism if and only if for any $g,h\in G$, we have $||\rho(g)\rho(h)-\rho(gh)||\leq \epsilon$ where the $||$ norm ...
7
votes
1answer
237 views

Division algebras over extension fields / reducibility of $G$-modules

Reformulation of the question (see below for the original question): Let $K$ be an algebraic number field and $D$ a finite-dimensional $K$-division algebra. Is there a description of the field ...
4
votes
1answer
143 views

Questions about the $\mathbf{i}$-trails of Berenstein and Zelevinsky

The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in this paper. It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^*$. Let ${\bf i}=(i_1, ...
2
votes
1answer
124 views

Why are compactly induced representations projective in the category of admissible representations?

I am reading part of Dipendra Prasad's paper found here: http://arxiv.org/pdf/1306.2729v1.pdf. In it (in the middle of page 8) he writes that compactly induced representations are projective. Why is ...
3
votes
1answer
246 views

Jones polynomial of tangles using Temperley-Lieb algbra?

The definition of the Jones polynomial of tangles (a la Reshetikhin and Turaev) uses the space of invariants for $U_q sl_2$ and R-matrices. It seems to me the same thing cane be done in terms of the ...
3
votes
1answer
244 views

Strong Morita equivalence and representation theory

In the context of pure algebra we say that two algebras (in general: rings) $A,B$ are Morita equivalence when there are bimodules $_AP_B,_BQ_A$ such that $P \otimes_B Q \cong _AA_A$ and $Q \otimes_A P ...
2
votes
1answer
225 views

Example of a Frobenius algebra that is not projective over a Frobenius subalgebra

I'd like to know an example of a Frobenius algebra $A$, with a subalgebra $B$ that is itself a Frobenius algebra, such that $A$ is not projective as a left $B$-module. I don't require any ...
1
vote
1answer
131 views

“Generators” for fusion rings

It's a rather obvious idea in the area of fusion rings, but I haven't found a reference yet. Start with the usual rules for a rank n fusion ring $X_i\bigotimes{X_j}=\Sigma_k{T_{ij}}^kX_k$ and ...
15
votes
2answers
410 views

ULU Decomposition of a matrix

Let $g \in GL_n(\mathbb{F}_q)$. Is it true that we can always write $g = u_1lu_2$, where $u_1$ and $u_2$ are upper-triangular and $l$ is lower-triangular? Note that I'm not requiring that the matrices ...
0
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0answers
63 views

“Reciprocal” of Schoenberg's theorem

Schoenberg's theorem states that for a (say, countable group) $G$ and any real valued conditionally negative type function $\psi$ on $G$, the function $e^{-t\psi}$ is positive definite, for any ...
6
votes
1answer
364 views

What is the universal property of quotienting a normaliser of the subgroup?

Let $G$ be a group, $H$ a subgroup and $X$ a $G$-set. By taking orbits $X/H = X \times_H 1$ or fixed points $X^H = \mathrm{Hom}_H(1,X)$ we obtain a set on which $H$ acts trivially, and we've destroyed ...
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0answers
130 views

Comparison of two Chevalley basis

Let $G$ be a connected reductive group over an algebraically closed field and $T$ a maximal torus. Let $H$ be a pseudo-Levi subgroup, say the neutral component of a centralizer of a semisimple element ...
5
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0answers
220 views

Torsors and twists of algebraic groups

Let $G/S$ be an affine group scheme. Then the automorphism group of every $G$-torsor over $S$ is a twist of $G$, but it this functor isn't essentially surjective in general (It may be not full nor ...
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0answers
65 views

kostant partition function vs Haar measure

I am trying to understand the relationship between the Kostant partition function and the Haar measure. Both seem to involve the Vandermonde determinant: $$ \Delta(\theta) = \prod_{i< j} ...
3
votes
1answer
206 views

Is there a nice description for the ring $\mathbb{C}[\mathfrak{g}]^G$, where $G$ is semisimple

Some context for my question. Given a vector space $V$, $\mathbb{C}[V]$ is defined to be the ring of polynomials on the vector space $V$. In other words, we can make the identification $\mathbb{C}[V] ...
3
votes
1answer
210 views

Representation of GL(n, F_p) over F_p, for n small

The question is related to this post Representation theory of the general linear group over a finite prime field However, I am asking for more detailed references for n small, for example, for n=2, ...
0
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0answers
92 views

Explicit basis/weight vectors for irreducibles inside the plethysm $Sym^m(\bigwedge^p \mathbf(V))$

This is a follow up to this question about finding the multiplicities of irreducible representations restricted to Young diagrams of 2-columns or less, inside the plethysm $Sym^m(\bigwedge^p ...
3
votes
2answers
173 views

counting points on nilpotent Springer fiber

Computing $p$-adic orbital integral I come to the following question. My ground field $k$ is the residue field of a non-arch local field, i.e. a finite field. I am happy to put any assumption on ...
2
votes
2answers
238 views

Closure relations between Bruhat cells on the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$. How do we prove the closure relations between the cells, which ...
4
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0answers
188 views

In general, are 'Young symmetrisers' given by Littlewood-Richardson 'Orthogonal projection Operators'?

Consider $V^{\otimes n}$ where $V$ is vector space and the representation of GL(V) acting in the usual way. Now if I consider tensor products or plethysms of irreducible spaces, this is not in general ...
2
votes
1answer
157 views

Relationship between Verma modules and delta functions

Reposting my question from math.stackexchange: What is the relationship between Verma modules and delta functions? Here's the quote from Woit's notes on Lie theory ...
3
votes
1answer
203 views

Base for symmetric group

Given symmetric group $S_n$, is it possible to find $k=\lceil\log_2S_n\rceil=\lceil\log_2n!\rceil$ members $\{\alpha_i\}_{i=1}^{k}$ in $S_n$ such that every member of $S_n$ can be written as ...
6
votes
1answer
182 views

What is the relation between spherical principal series representations of a reductive Liegroup and Verma modules for its Lie algebra?

Here is an issue that thoroughly confuses me. I hope I can express it in a way that is clear cut enough for this site. Let $G$ be a real reductive Lie group and $\mathfrak{g}$ be the complexification ...
-1
votes
1answer
167 views

A 6j multiplicity paradox

I somewhat advanced since 6j symbols trouble when irrep multiplicity >1 (because I found the decades old papers where the Racah algebra is dealt with, e.g. "Coupling Coefficients and Tensor ...
0
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0answers
58 views

Computer software for manipulating loop groups or matrices with polynomial entries

I need to deal with loop groups $LG$ over the complex numbers $\mathbb{C}$, as well as related spaces like the affine Grassmannian and affine flag variety a lot. In type A, the loop group consists ...
1
vote
2answers
176 views

How to find the multiplicity of weight in a Verma module?

In particular, let $\mathfrak g$ be the semisimple Lie algebra of type $A_{2}$ et let $\alpha,\beta$ be its simple roots. How can the multiplicity of weight $-2\alpha -3\beta$ be calculated in the ...
4
votes
2answers
117 views

perturbation of Invariant subspaces

Let $A,B$ be matrices in $GL(n,\mathbb{R})$ sufficiently close in the usual metric on matrices. Suppose $A$, resp., $B$ stabilizes a $k$-dimensional subspace $U$, resp., $V$ of $\mathbb{R}^n$, where ...
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0answers
48 views

abstract affine representations of semisimple Lie groups

in 1933 van der Waerden proved that any abstract unitary representation of a compact semisimple Lie group is necessarily continuous. Is any kind of similar result known for abstract affine ...
5
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1answer
104 views

How can the generators of subalgebra $\mathfrak g^{\sigma}$ of $\sigma$-stable elements be expressed through generators of Lie algebra $\mathfrak g$?

Let $\mathfrak g$ be the semisimple Lie algebra of type $D_{4}$. Let $\sigma$ be the 3-rd order automorphism of $\mathfrak g$ induced by the triality of $D_{4}$: $$ ...
1
vote
1answer
170 views

What is the cubic casimir element of sl_3?

I have been thinking about this for some time but have had no luck. I have found some sources that say higher Casimir elements can be obtained by generalizing the second order Casimir, which is ...
3
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0answers
91 views

Newvectors in tensor product representations

Let $\pi_p$ and $\pi_p^\prime$ two smooth admissible irreducible complex representations of ${\rm GL}_2(F)$ where $F$ is a non archimedean local field of residual characteristic $p$ of central ...
1
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1answer
162 views

Can we say that $A$ is a complement for a group $G$?

Let $A$ be a frobenius complement for a group $G$ i.e. $A$ act on $G$ by automorphism s.t. $C_A(g)=e$ for all nonidentity $g$. Now, Action of $A$ can be linearly extended so that $A$ act on $F[G]$. ...
9
votes
1answer
183 views

Powers of traces, integrals over spheres and class functions

I asked this on math.StackExchange a while back but got no answers. I hope I'll be forgiven for the double post. Let $V$ be a complex vector space of dimension $\operatorname{dim}_{\mathbb C} V = n$, ...
5
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0answers
184 views

Unitary representations of Tarski Monsters and other beasts

Did people study the unitary representations of Tarsky Monsters, for example the ones constructed by Ol'shanskii? Are there any exotic representations, ie. except the ones related to the left regular, ...
1
vote
1answer
93 views

Reducible reductive Lie subalgebras of so(p,q)

Is it true that $S(O(p) \times O(q))$ is the only proper subgroup of $SO(p,q)$ of full rank acting on the natural representation $\mathbb{R}^{p+q}$ of $SO(p,q)$ that stabilizes a $p$-dimensional ...
12
votes
2answers
349 views

Langlands duality and multiplying cocharacters

Recall that there is a bijection between irreducible representations of a compact real Lie group $G$ and the cocharacters (homomorphisms $U(1) \to G$, modulo conjugation) of the Langlands dual group ...
1
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1answer
188 views

A question on the representation theory of finite group

By the Burnside theorem, we know that we can decompose the order of a group in to a sum of some integers' square, and these integers are the dimensions of the group's irreducible representations . ...
0
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1answer
81 views

Reductive subgroup and its derived subgroup with an irreducible represenation

Could you please answer the following question: Let $V$ be a faithful irreducible representation of a connected reductive group $H$ defined over $\mathbb{R}$ Is it true that the derived group of $H$, ...
2
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0answers
118 views

Irreducible representations of $Sp(4,\mathbb{F}_2)$

I'm trying to construct the irreducible representations (over $\mathbb{C}$) of the finite group $Sp(4,\mathbb{F}_2)$. Using GAP, the character table is as follows: $$ \left(\begin{matrix} 1 & 1 ...
4
votes
4answers
301 views

Breaking up the free Lie algebra into Gl irreps

The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of https://en.wikipedia.org/wiki/Free_Lie_algebra , the free Lie algebra generated by any choice of ...