All Questions
Tagged with reference-request rt.representation-theory
823 questions
10
votes
1
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642
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Is the affine closure of the basic affine space of a reductive algebraic group Cohen–Macaulay?
Let $G$ be a reductive algebraic group with choices of Borel subgroup and maximal torus $B \supseteq T$ and unipotent radical $U$ over an algebraically closed field $k$ of characteristic zero. Is the ...
- 422
4
votes
1
answer
321
views
Reference for fact about reduction mod $p$ of a representation of a finite group
Let $G$ be a finite group, let $M$ be a $\mathbb{Z}[G]$-module whose underlying abelian group is finite-rank free abelian, and let $\mathbb{F}$ be a field of characteristic $0$. For some prime $p$ ...
- 41
2
votes
0
answers
91
views
When does a stable endomorphism ring have injective dimension at most one?
tLet $A$ be a Frobenius algebra (we can assume that $A$ is given by quiver and relations) and let $M$ be a basic $A$-module without projective direct summands (we can assume we know the decomposition ...
- 26.5k
11
votes
1
answer
617
views
Is there a unique "natural" action of $\mathsf{SL}_{n+1}$ on $\mathbb{R}^n$?
Context
By acting naturally via $\mathsf{SL}_3$ on $\mathbb{RP}^2=\{[x:y:z]\}$ and by taking the induced action on the affine hyperplane $z=1$ (which we identify with $\mathbb{R}^2$), one can realize ...
- 2,527
6
votes
1
answer
226
views
Tensor representations of the quantum algebra $U_q(\mathfrak{sl}(2))$ at the roots of unity
I'm trying to understand how the representation theory of $U_q(\mathfrak{sl}(2))$
works and I had a look to some books and lecture notes available on the internet. The case of $q^m\neq1$ is discussed ...
2
votes
1
answer
75
views
Reference for the action of the Mullineux involution on a partition with an added good node
Could you help me to locate a reference in the literature to the following fact: if you act by the Mullineux involution $M$ on a partition $Q$ which is a union of a regular partition $P$ with an added ...
4
votes
1
answer
670
views
Formula involving Wigner's 3j symbols and integration over irreducible representations of SU(2)
$\DeclareMathOperator\SU{SU}$In some calculations, I saw the following formula
$$\int_{\SU(2)}\,\mathrm{d}g\,D^{j_{1}}_{m_{1}n_{1}}(g)D^{j_{2}}_{m_{2}n_{2}}(g)D^{j_{3}}_{m_{3}n_{3}}(g)=(-1)^{j_{1}+j_{...
- 1,171
4
votes
0
answers
163
views
External tensor product Calabi-Yau DG categories
Let $\mathcal{C}$ be a smooth proper DG-category such that the shift $[p]$ is a Serre functor for $D^{perf}(\mathcal{C})$ (we say that $D^{perf}(\mathcal{C})$ is $p$-Calabi-Yau). I am looking for a ...
- 7,300
2
votes
0
answers
382
views
Is there a roadmap to learning representation theory of finite group over finite field?
I've been wanted to learn some basic theories of the (non-semisimple) representation of the finite group over a finite field.
I have been guessing that the materials might be contained in the books on ...
- 1,013
7
votes
3
answers
599
views
Root system of fixed point Lie sub-algebra
It is known that a non-simply laced simple root system can be constructed from the simply-laced root system by folding the Dynkin diagram and hence the corresponding non-simply-laced Lie algebra can ...
- 103
2
votes
1
answer
317
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English translation of Emmy Noether's Hyperkomplexe Grössen und Darstellungstheorie
I'm wondering if anybody knows where one can find an English translation of Emmy Noether's classical paper E. NOETHER, Hyperkomplexe Grössen und Darstellungstheorie, Math. Zeit. 30(1929), 641–692 ?...
- 38.6k
1
vote
1
answer
275
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The norm of the principal series intertwining operator for $\operatorname{GL}_2$
Is there a known bound on the norm of the standard intertwining operator for the principal series of $G = \operatorname{GL}_2(\mathbb Q_p)$?
Background:
For a character $\chi = (\chi_1,\chi_2)$ of the ...
- 6,180
3
votes
2
answers
448
views
Is there any simple formula for the character of $S_{n}$ represented by the set of $k$-tuples of $\{1,2,...,n\}$?
I'm interested in the representation theory of symmetric groups.
I'm now trying to search for the formula for the characters of $\Omega^{k}$, the set of $k$-tuple of elements of $\Omega$ a set of $n$ ...
- 1,013
10
votes
7
answers
2k
views
Representations of products of symmetric groups
I'm writing a paper and want to cite some references to efficiently prove that over any field $k$ of characteristic zero, every irreducible representation of a product of symmetric groups, say
$$ S_{...
- 22.3k
3
votes
1
answer
305
views
What corresponds to the operation of taking traces in of the Fourier transformation on a finite group?
I have a question about the Fourier transfomation on a finite non-comutative group. I hope that it is a known fact in the Representation Theory but I cannot find it written explicitly in textbooks.
...
- 41.8k
8
votes
1
answer
264
views
Class group of hypersurfaces of finite representation type
Let $k$ be an algebraically closed field of characteristic different from $2,3$ and $5$, and let $R=k[[x,y,x_2,\dots,x_d]]/(f)$, where $f\in(x,y,x_2,\dots,x_d)^2$, $f\neq0$. By results of Buchweitz-...
- 411
9
votes
1
answer
355
views
Reference request for indecomposable representations of $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$
Is there a good reference on the classification of indecomposable representations of the Lie algebra $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$ (of course, in ...
- 239
2
votes
1
answer
212
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Changing $S_2 \wr S_n$ for $S_n \wr S_2$ in the theory of zonal polynomials
The permutation group $S_{2n}$ has $H_{2,n}=S_2\wr S_n$ as a subgroup. The plethysm $h_n(h_2)=\sum_{\lambda\vdash n}s_{2\lambda}$ is well known.
The zonal spherical functions $\omega_\lambda(g)=\frac{...
- 1,549
8
votes
0
answers
481
views
Formal degree of discrete series representations
Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is said to be a discrete series if its matrix coefficients (defined by $\xi^\pi_{v,w} : g \...
- 5,424
1
vote
0
answers
90
views
Dimension of a certain space of symmetric functions: Part II
This is the second installment of my earlier MO question.
Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. Denote the set of all partitions with distinct parts by $\...
- 43.2k
4
votes
0
answers
205
views
Dimension of a certain space of symmetric functions: Part I
Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. A partition $\lambda$ is called a $t$-core if none of its hook lengths are multiples of $t$.
QUESTION. Consider the ...
- 43.2k
2
votes
0
answers
104
views
$G$-module representations of a profinite quiver
I have a profinite directed graph $\Gamma$, i.e., I can think of $\Gamma$ as the inverse limit of a directed system of finite directed graphs under inclusion. To each vertex of the graph a $G$-module ...
- 21
2
votes
1
answer
316
views
Decomposition of Hilbert spaces via groups and algebras representations
Let $\mathcal{H}$ be a complex finite dimensional Hilbert space and let $\mathcal{A}\subseteq \mathcal{B}(\mathcal{H})$. I am looking to understand the different decompositions of $\mathcal{H}$ ...
- 23
3
votes
0
answers
115
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Reference for the Netto's theorem on the permutation groups which was mentioned in the paper of Frobenius
I'm trying to read 'Uber die Charaktere der mehrfach transitiven Gruppen' written by Frobenius.
There he mentioned some theorems of Netto.
I'm depending on the Google translator. and the translation ...
- 1,013
8
votes
2
answers
482
views
Parabolics and simple roots for a special unitary group: reference request
I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed.
...
- 14.1k
2
votes
0
answers
84
views
symmetry for a pair of statistics on partitions
Let $\lambda\vdash n$ denote a partition $\lambda$ of $n$ and let $\square\in\lambda$ denote a box $\square$ in the Young diagram of $\lambda$.
QUESTION. Can you list a pair of (distinct) statistics $...
- 43.2k
6
votes
1
answer
252
views
Is there a known classification of regular multiplicity-free permutation groups?
The question is in the title, but let me clarify the terminology. I consider a permutation group $\Sigma\subseteq\mathrm{Sym}(\Omega)$ on a finite set $\Omega$.
$\Sigma$ is regular if it acts ...
- 13.6k
6
votes
1
answer
215
views
What is the smallest group for which Broué's abelian defect group conjecture has not yet been verified?
Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $k:=\overline{\mathbb{F}_p}$.
Let $b$ be a $p$-block of $kG$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer ...
- 1,755
1
vote
0
answers
101
views
Reference request for the representation of Metaplectic group and related topics
I've studied the spin geometry, Atiyah-Singer index theorem and I realize that the representation theory of spin groups, and spinor representation is very interesting and useful things in geometry.
...
- 285
18
votes
1
answer
458
views
For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$?
Title. For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$?
If $G$ is a subgroup of either $S^0,S^1,S^3$ or $S^7$ this induces a free action ...
- 363
0
votes
0
answers
181
views
Request for a modern Reference for Frobenius' paper "Über die Charaktere der mehrfach transitiven Gruppen"
I'm interested in the paper of Jan Saxl "The Complex Characters of the Symmetric Groups that Remain Irreducible in Subgroups".
I have only (not yet enough!) standard background on the ...
- 1,013
4
votes
1
answer
104
views
Which Auslander algebras satisfy $Ext_B^1(D(B),B)=0$?
Let $B$ be the Auslander algebra of a representation-finite algebra $A$.
Question: When do we have $Ext_B^1(D(B),B)=0$? Can this be expressed in terms of nice properties of $A$?
This is for example ...
- 26.5k
5
votes
0
answers
149
views
Quiver and relations of $F\mathrm{SL}(2,q)$
$\DeclareMathOperator\SL{SL}$Let $q=p^n$ be a prime power and $F$ a field of characteristic two.
Let $G=SL(2,q)$ the group of $2 \times 2$ special linear matrices over the field with $q$ elements with ...
- 26.5k
4
votes
0
answers
107
views
Which algebras of quaternion type do appear as blocks of group algebras?
A finite dimensional algebra $A$ is said to be of quaternion type (first defined by Erdmann?) when it is tame, symmetric and connected and furthermore the Cartan matrix is nonsingular and the stable ...
- 26.5k
6
votes
2
answers
331
views
Lie powers of a graded vector space and Klyachko's theorem
Let $V$ be a $\mathbb{Z}_2$-graded vector space (aka super vector space) and $L(V)$ be the free $\mathbb{Z}_2$-graded Lie algebra (aka super Lie algebra). The free super Lie algebra is also graded by ...
- 21.5k
10
votes
1
answer
307
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Rings where all indecomposable projective modules are finitely generated
Let $X$ be the class of (unital, associative and not necessarily commutative) rings $R$ where every indecomposable projective $R$-module is finitely generated.
Question 1: Is there a nice equivalent ...
- 26.5k
6
votes
1
answer
294
views
Rickard's strengthening of Broué's abelian defect group conjecture and the lifting of some equivalences up to splendid derived equivalences
Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $K:=\overline{\mathbb{F}_p}$.
Let $b$ be a $p$-block of $G$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer ...
- 1,755
3
votes
3
answers
356
views
Absolutely irreducible finite reflection/rotation groups
Suppose that $G$ is a finite irreducible reflection group with irreducible orthogonal representation $\rho: G\rightarrow \mathrm{O}(d)$, and let $\rho^+: G^+\rightarrow \mathrm{SO}(d)$ be its ...
- 439
1
vote
0
answers
213
views
Is there any research on the action of a subgroup on the whole finite group by conjugation?
I want to know whether there are any research on the orbits of the action of a subgroup by conjugation on the whole group, when the group is finite. (Especially whole symmetric group.)
I'm especially ...
- 1,013
0
votes
2
answers
283
views
Motivation and reference for Brauer algebras
I am looking for a good reference and motivation for Brauer monoid and Brauer algebras. Kindly help me with some suggestions. Thanks.
- 141
1
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0
answers
142
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Principal orbit and the generic stabilizer of SO(2n)xSO(2n)
Let $SO(2n)$ denote the special orthogonal group of $2n\times 2n$ matrices over the complex numbers.
Consider the action of $SO(2n)\times SO(2n)$ on the set of $2n\times 2n$ matrices : $ADB^{T}$, ...
- 11
1
vote
0
answers
89
views
Is there any English reference for the paper 'Darstellungstheorie von Schur-Algebren' written by Fredrich Roesler?
Now I'm reading the paper of Friedrich Roesler on the representation theory of Schur-Rings with the title 'Darstellungstheorie von Schur-Algebren' (Math Z 1972).
My goal is to understand algebraic ...
- 1,013
9
votes
1
answer
248
views
Decomposition of $\bigotimes^{m} \mathbb{C}^{n}$ under the action of $\operatorname{GL}_{n}\times \operatorname{S}_{m}$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\S{S}$I want to know the proof of the following theorem. It is stated somewhere that, a proof can be found in: "Roger Howe, Perspectives on ...
- 179
5
votes
0
answers
142
views
A practical way to check whether a module is periodic
A module $M$ over a finite dimensional selfinjective algebra $A$ over a field $K$ is called periodic if $M \cong \Omega^n(M)$ for some $n \geq 1$. We assume here that $M$ is simple and that A is a ...
- 26.5k
13
votes
0
answers
237
views
A Dynkin type classification result in linear algebra
Let $G$ be a finite directed acyclic graph. The Cartan matrix $C_G=C$ of $G$ is defined as the matrix with rows and colums indexed by the vertices of $G$ and $c_{i,j}$ counts the number of paths from $...
- 26.5k
10
votes
1
answer
299
views
Map from Bruhat stratification to Catalan stratification for the space of totally nonnegative upper-triangular matrices
$\DeclareMathOperator\SL{SL}$This question came up in a class "Total Positivity and Cluster Algebras" being taught by Chris Fraser.
Let $N^+$ denote the space of uni-upper-triangular ...
- 24.2k
5
votes
1
answer
346
views
Restricting representations to a principal $\mathfrak{sl}(2)$
Let $\mathfrak{g}$ be a semi-simple Lie algebra over $\mathbb{C}$ with simply connected group $G$ and suppose that
$$\mathfrak{g} = \bigoplus_i\mathfrak{g}_i$$ is a $\mathbb{Z}$- or $\mathbb{Z}/n\...
- 2,516
3
votes
1
answer
473
views
Borel–Weil–Bott for partial flag varieties
Is there a generalization of Borel-Weil-Bott for partial flag varieties, i.e. homogeneous spaces of the form $G/P$ with $P$ parabolic and $G$ semisimple? If so, I would like a reference.
- 3,079
14
votes
2
answers
897
views
A canonical bijection from linear independent vectors to parking functions
Call an $n$-vector $v$ in $\mathbb{Z}^n$ cool when it has only entries 0 or 1 and the ones appear in only one block. Thus there are $n(n+1)/2$ such vectors. For $n=3$ they are:
[ <[ 1, 0, 0 ]>, &...
- 26.5k
6
votes
1
answer
657
views
Does the purported proof of Rota's conjecture provide an algorithm for calculating the forbidden minors of matroids over arbitrary finite fields?
About six years ago there was a proof announced and later outlined in a notice from AMS. However right now I can only seem to find forbidden minor characterizations for matroids linearly ...
- 2,506