Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
6,803
questions
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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 ...
0
votes
0
answers
173
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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 ...
8
votes
1
answer
479
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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 ...
13
votes
3
answers
1k
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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
1
answer
170
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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 ...
5
votes
2
answers
206
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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, ...
2
votes
1
answer
117
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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 ...
9
votes
1
answer
510
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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 ...
6
votes
1
answer
252
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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, \...
3
votes
1
answer
667
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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 ...
4
votes
1
answer
864
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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 ...
8
votes
1
answer
714
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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
1
answer
398
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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 ...
2
votes
1
answer
211
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"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 ...
16
votes
2
answers
730
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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 ...
10
votes
1
answer
459
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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 ...
7
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0
answers
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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 ...
1
vote
0
answers
160
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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} |e^{i\...
3
votes
1
answer
276
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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
1
answer
549
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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, ...
1
vote
0
answers
214
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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 \mathbf(V)...
3
votes
2
answers
584
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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 $\...
3
votes
2
answers
658
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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 ...
8
votes
2
answers
1k
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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
1
answer
366
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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 (http://www.math.columbia.edu/~woit/...
3
votes
1
answer
243
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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 $\alpha_1^...
14
votes
2
answers
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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 ...
0
votes
1
answer
260
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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 ...
3
votes
2
answers
666
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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
2
answers
206
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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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0
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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
votes
1
answer
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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}$:
$$
\sigma:\alpha_{1}\mapsto\alpha_{3}\...
5
votes
2
answers
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What is the cubic Casimir element of $\mathfrak{sl}_3$?
$\DeclareMathOperator\ad{ad}\DeclareMathOperator\Tr{Tr}$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 ...
3
votes
0
answers
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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
vote
1
answer
173
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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 $F[G]$. ...
10
votes
1
answer
357
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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
votes
0
answers
312
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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
1
answer
163
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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 ...
13
votes
2
answers
602
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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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1
answer
207
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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
votes
1
answer
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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
votes
0
answers
356
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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 ...
12
votes
4
answers
800
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Breaking up the free Lie algebra into GL irreps
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\L{\mathfrak{L}}$The free Lie algebra $\L(V)$ generated by an $r$-dimensional vector space $V$ is, in the
language of https://en.wikipedia.org/wiki/...
2
votes
0
answers
303
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Rational conjugation of elements of a finite group
Let $G$ be a finite group. Two elements $x$ and $y$ of $G$ are said to be rationally conjugate, written $x \sim_{r} y$, if and only if $\langle x\rangle$ and $\langle y\rangle$ are conjugate subgroups ...
4
votes
1
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194
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Decomposing representations of finite groups of Lie type via computer
This is related to my previous question here.
Let me remind you what that question asked:
Let $\text{St}_n(\mathbb{F}_q)$ be the Steinberg module (over $\mathbb{C}$) for $\text{SL}_n(\mathbb{F}_q)$.
...
2
votes
0
answers
268
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Functoriality for non-split orthogonal groups
I am trying to understand the functoriality conjectures of Langlands. We know that the functoriality conjectures imply that automorphic $L$-functions of a connected reductive group are equal to ...
7
votes
0
answers
489
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Can one classify irreducible unitary representations of the Weyl algebra?
I saw in this MO post: Is there a machinery describing all the irreducible representations ? that classifying irreducible representations of the Weyl algebra is essentially intractable. My question is ...
11
votes
2
answers
2k
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Representation theory of the general linear group over a finite prime field
I am re-posting a question I asked on math.se here because I am unsatisfied with the answers I obtained.
The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely ...
1
vote
0
answers
120
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Computing $\text{Tor}_*^{R_G} (\mathbb{Z}, \mathbb{Z}) $ for a compact Lie group $G$
Let $R_G$ be the representation ring of $G$ a connected, simply connected Lie group, $I_G$ the augmentation ideal and $\mathbb{Z}=R_G/I_G$. $R_G$ acts on $\mathbb{Z}$ via $V \cdot n = (\dim V ) n$. I ...
2
votes
0
answers
141
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About the reduceness of the commuting scheme associated with a symmetric pair
my question is the following one:
Let $G$ be a connected reductive algebraic group over the field of complex numbers, and let $V$ be a linear representation of $G$ obtained as the isotropy ...