All Questions
Tagged with rt.representation-theory reference-request
823 questions
7
votes
2
answers
301
views
Reference for projective covers of direct products of finite groups?
This concerns one of those "well known" facts, referred to in a recent preprint I've been looking at. In principle it's elementary, but I can't pin down an explicit textbook reference for it. ...
- 2,821
35
votes
6
answers
5k
views
Character-free proof that Frobenius kernel is a normal subgroup?
The question is in the title, but here is some background/reminders:
A subgroup $H\neq\{1\}$ of a finite group $G$ is called a Frobenius complement if $H\cap H^g = \{1\}$ for all $g\in G\backslash H$....
- 63.9k
12
votes
1
answer
567
views
Reference for character sheaves over $\mathrm{GL}_n(q)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$I know a little bit about complex representation theory of finite reductive groups as $\GL_n(q),\SO_n(q)$ etc via Deligne-Lusztig induction and ...
- 437
5
votes
1
answer
1k
views
The Casimir invariant of an irreducible representation of a compact Lie group
Let $G$ be a compact Lie group (not necessarily connected) and $\rho:G\to \mathrm{End}(V)$ an irreducible (hence finite-dimensional) unitary representation of $G$. Let $\mathfrak{g}$ be the Lie ...
- 63.9k
4
votes
1
answer
520
views
List of Casimir elements of low dimensional Lie algebras
I am interested in explicit formulae for the Casimir elements (or "Casimir operators") of low-dimensional, real, non-Abelian Lie algebras (d=2,3, and possibly 4). I am wondering if there is any ...
- 63.9k
2
votes
2
answers
680
views
Complete representation theory of $\mathrm{SL}(2,\mathbb R)$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is the complete representation theory of $\SL(2,\mathbb R)$, $\GL(2,\mathbb R)$, $\SL(2,\mathbb C)$, and $\GL(2,\mathbb C)$ known, in the sense ...
- 954
2
votes
1
answer
214
views
A Vandermonde like determinant with exponentials
Let $n\geq m$ be non negative integers, and consider a list of $(n+m+1)$ distinct numbers (complex or real). I am interested in getting a closed form formula for the following determinant: $\det\left[\...
- 3,691
9
votes
3
answers
3k
views
About representation theory of Heisenberg group
Actually I am an undergraduate student, but I want to study Heisenberg groups over arbitrary field.
Firstly, why is this group important? I know that the Heisenberg group is important in the field ...
- 12.9k
2
votes
0
answers
138
views
Construction of a certain long exact sequence
Let $A$ be a noetherian ring (not necessarily commutative) or for simplicity even a finite dimensional algebra over a field.
Let $X$ and $U$ be finitely generated $A$-modules and let $add(U)$ be the ...
- 26.5k
1
vote
1
answer
1k
views
Does it make sense that "Representations of groups over finite ring" ?
I am an undergrad student who wants to know about the representation theory over
arbitrary finite fields or finite rings of characteristic p (p a prime). (called modular
representation theory.)
In ...
- 1,446
10
votes
1
answer
399
views
Basic algebra of $\mathcal{O}_0(\mathfrak{sl}_n(\mathbb{C}))$ — Reference request
It is well known that the principal block $\mathcal{O}_0$ of the BGG category $\mathcal{O}$ of a semisimple Lie algebra is equivalent to the category of finitely generated modules over a certain ...
- 26.5k
3
votes
0
answers
463
views
Representations of triangle groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up.
Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...
- 63.9k
13
votes
1
answer
853
views
Applications of equivariant homotopy theory to representation theory
Equivariant homotopy theory focuses on spaces together with some group action on them. Jeroen van der Meer and Richard Wong have a paper where they use equivariant methods to compute the Picard group ...
- 11.1k
2
votes
0
answers
98
views
Question concerning relationships between different $p$-modular systems and Brauer character values
Let $(K,\mathcal{O},k)$ be a large enough $p$-modular system, where $\mathcal{O}$ is a complete discrete valuation ring of characteristic zero with unique maximal ideal $J(\mathcal{O})$, algebraically ...
- 1,755
10
votes
1
answer
642
views
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
273
views
Wedderburn decomposition of special linear groups
$\DeclareMathOperator\SL{SL}\newcommand\card[1]{\lvert#1\rvert}$I want to study about Wedderburn decomposition of group algebra $k\SL(n,\mathbb{F}_p)$ where $k$ is either an algebraically closed field ...
- 13k
18
votes
2
answers
1k
views
Has anyone seen this construction of the Weil representation of $\mathrm{Sp}_{2k}(\mathbb{F}_p)$?
$\def\FF{\mathbb{F}}\def\CC{\mathbb{C}}\def\QQ{\mathbb{Q}}\def\Sp{\text{Sp}}\def\SL{\text{SL}}\def\GL{\text{GL}}\def\PGL{\text{PGL}}$Let $p$ be an odd prime. The Weil representation is a $p^k$-...
- 44.4k
7
votes
1
answer
185
views
Origin of the abacus bijection
What is the origin of the abacus bijection (aka the rim hook bijection, aka the Stanton-White bijection, aka James's bijection)?
Igor Pak, in his 2000 article "Ribbon tile invariants" (...
- 19.7k
3
votes
1
answer
106
views
Reference request for equivalent formulations of being absolutely indecomposable
I would like to ask the following question.
I am searching for a reference for the following statement:
Suppose $k$ is a perfect field. Let $A$ be a (symmetric) $k$-algebra and let $M$ be a finitely ...
- 311
13
votes
0
answers
615
views
The derived category of integral representations of a Dynkin quiver
Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write $\...
- 2,821
5
votes
1
answer
61
views
Do $F$-traces of simple modules at $p'$-classes uniquely determine the module?
Let $G$ be a finite group and let $p$ be a prime number such that $p\mid |G|$.
Let $\text{IBr}(G)$ denote the set of irreducible Brauer characters of $G$ for the prime $p$.
Assume $\mathbb{F}_{q}$ is ...
- 44.4k
18
votes
0
answers
895
views
local equivalence of loop group representations
Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the local loop group
$$
L_IG ...
- 2,821
2
votes
0
answers
92
views
Is there a standard reference for: taking projective covers of simple modules commutes with finite Galois field extensions?
Let $\Lambda$ be an Artin algebra over a finite field $k$. Is there a standard reference for the fact that taking projective covers of simple $\Lambda$-modules commutes with finite Galois field ...
- 311
12
votes
2
answers
926
views
Finite groups with integral character table
The character table of a finite group will be called integral if all its entries are integers. There are $11$ such groups up to order $16$, namely $C_1$, $C_2$, $C_2^2$, $S_3$, $D_8$, $Q_8$, $C_2^3$, $...
- 44.4k
2
votes
0
answers
84
views
Representation finite Hopf algebras up to stable equivalence
It is well known that every representation-finite group algebra $KG$ is stable equivalent to a symmetric Nakayama algebra.
Question: Is it true that every representation-finite Hopf algebra is stable ...
- 66
1
vote
0
answers
60
views
Alternative definition of physical states
Suppose that we have a vertex operator algebra $V$ with a conformal element $\omega$ and the associated conformal field
$$
Y(\omega,z) = \sum_{k\in \mathbb{Z}} L_kz^{-k-2}\,,
$$
where $L_k$ satisfy ...
- 988
4
votes
0
answers
128
views
Filtrations of the irreducible representations of the symmetric groups
For a partition $\lambda$ of $n$, write $S^{\lambda}$ for the irreducible $S_n$-representation that corresponds to $\lambda$ (a.k.a. the Specht module).
For two integers $d<n$ write $Par_d(n) = \{\...
- 5,039
5
votes
1
answer
362
views
Reference for Mackey functors with group value inverted
I'm looking for a reference for the decomposition of the category of Mackey functors for a finite group when the order of the group is inverted. (There is also an analogous decomposition for the ...
- 3,784
1
vote
0
answers
97
views
Minimizing distance over finite group action
Let $G$ be a finite group and $V$ a unitary irreducible rep’n of dimension $N$. Is there a fast (polynomial in $\log|G|$) algorithm to compute $\displaystyle \min_{g \in G}d(x,gy)=\max_{g \in G} Re\...
- 571
16
votes
3
answers
1k
views
Reference for representation theory of SL_2(Z/n)
There are many references for the representation theory (say over $\mathbf C$) of $\operatorname{SL}_2(\mathbf{F}_q)$ and $\operatorname{GL}_2(\mathbf{F}_q)$, for instance lecture 5 in Fulton--Harris &...
10
votes
0
answers
234
views
Branching from GL(a+b) to GL(a) x GL(b)$ using Gel'fand-Cetlin patterns
If one iterates the multiplicity-free branching rule from $\operatorname{GL}(n)$ representations (finite-dim, over $\mathbb C$) to $\operatorname{GL}(n-1)$ all the way down to $\operatorname{GL}(0)$, ...
9
votes
1
answer
493
views
A compactification of the space of points on the affine line
I recently encountered an interesting space. It is a compactification of the space of $ n$ points in $ \mathbb A^1 $ modulo translation, $ (\mathbb A^1)^n / \mathbb G_a $.
Let $ n \in \mathbb N $ and ...
- 2,537
9
votes
0
answers
210
views
Why and how is a representation "continuously decomposable"?
What I am asking may apply to a much more general setting and I am interested in the underlying level of generality of these statements, mostly with canonical references. However I state the question ...
- 5,424
6
votes
0
answers
365
views
Is this just a numerical accident or what?
In a complementary proof for a matrix determinant of $a_{i,j}=\binom{n-1+i}j$, raised by BillyJoe, I showed the more general evaluation
$$\det\left(\binom{i+p}{j+k-1}\right)_{1\leq i,j\leq m}
=\prod_{...
12
votes
2
answers
701
views
Character theory and Quantum Chemistry
Who (presumably a chemist) realized first the efficiency of character theory in calculations of orbitals of atoms? In which year?
- 63.9k
2
votes
0
answers
47
views
Character of the Young product of representations of a Lie group
For a compact (reductive/semisimple) Lie group $G$ with a maximal torus $T$, which I will identify with a subgroup of ${\mathbb{C}^*}^n$ (for simplicity let's just say that $G\leqslant GL(n,\mathbb{C})...
- 3,186
5
votes
1
answer
207
views
Finite lattices that are Koszul
Let $L$ be a finite lattice and $A=KL$ the incidence algebra of $L$.
It should be true that $L$ is modular if and only if the algebra $KL$ is quadratic (since being modular is equivalent to having no ...
- 26.5k
5
votes
1
answer
792
views
Constructing a Kac-Moody group as a quotient of the free product of its root subgroups
The paper "Regular Functions on Certain Infinite-dimensional Groups" by Kac and Peterson describes the construction of a group associated to the datum of a Kac-Moody algebra in a way I haven'...
- 63.9k
4
votes
0
answers
163
views
An identity for Schur polynomials
Given a partition $\lambda$, the Schur polynomials can be defined, among many other ways, as
$$S_{\lambda}(\xi_1,\dots,\xi_a)=\frac{\det\left(\xi_i^{\lambda_j+a-j}\right)_{i,j=1}^a}{\det\left(\xi_i^{a-...
- 43.2k
2
votes
2
answers
230
views
Does the map $f \mapsto \mu_f$ (BV to Lebesgue-Stieltjes measure) behave nicely under function concatenation?
Consider two continuous functions $f,g : [0,1]\rightarrow\mathbb{R}$ of bounded variation, and let $\mu_f, \mu_g : \mathcal{B}([0,1])\rightarrow\mathbb{R}$ be their associated Lebesgue-Stieltjes (...
- 12.9k
4
votes
1
answer
108
views
How much is known about the non-degeneracy of Quiver-with-potential associated to closed punctured surfaces?
The potential of the quiver associated to surfaces is the canonical one given by Labardini-Fragoso's 2009 paper, who proved that the the QP associated to surfaces whose boundary is nonempty is rigid ...
- 7,875
2
votes
0
answers
70
views
Rigid modules for hereditary algebras
Let $A=KQ$ be a path algebra of a connected quiver. (K algebraically closed if it helps)
Question: Is there an explicit classification of all indecomposable $A$-modules $M$ that are rigid, that is $...
- 26.5k
5
votes
0
answers
263
views
Reference/list of reductive subgroups of reductive groups?
Let $G$ be a (say, connected) reductive group over an algebraically closed field of characteristic zero (say, $\mathbb C$).
I am looking for simple examples of (ideally) complete characterizations of ...
- 3,799
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 ...
CommunityBot
- 1
2
votes
0
answers
87
views
Reference request on Plancherel measure for partitions whose parts differing by more than $1$
Given an (unrestricted) integer partition $\lambda$ of $n$, let $f_{\lambda}$ denote the number of standard Young tableaux (SYT) of shape the Young diagram $Y(\lambda)$ of $\lambda$. Then,
$$\sum_{\...
- 43.2k
7
votes
1
answer
227
views
Invariants for the isotropy representation of a Riemannian symmetric space
Statement: Let $M = G/K$ be a Riemannian symmetric space of compact type, and $V = T_o M$ be its isotropy representation (of $K$ acting on the tangent space of $M$). Then the Hilbert–Poincaré series $...
- 108k
4
votes
1
answer
155
views
Resource on spectral theory for differential operators with symmetry groups
In Methods of Mathematical Physics IV by Reed and Simon, the authors cover Floquet theory in detail in Section XIII.16. On page 280, they note that
"A part of the analysis of [the periodic ...
- 133
1
vote
0
answers
287
views
Characters of upper triangular matrices over finite field - reference request
Let $B_n$ be the group of upper matrices and $U_n$ the subgroup of unipotent upper triangular matrices. I would like some references which discusses complex character theory of $B_n(\mathbb{F}_q)$ for ...
- 2,751
5
votes
2
answers
1k
views
What is the motivation behind symplectic/orthogonal content?
Here $\lambda'$ is the conjugate partition of $\lambda=(\lambda_1,\lambda_2,\dots)$ and cells are in the Young diagram.
The symplectic content of cell $(i,j)$ of $\lambda$ is defined by
$$c_{sp}(i,j)=\...
- 25.2k
4
votes
1
answer
565
views
Connection between Deligne-Mostow monodromy and Gassner representation at roots of unity of the pure braid group
I am looking for a specific reference to the connection between [1] the Deligne-Mostow monodromy and [2] Gassner representation at roots of unity of the pure braid group. I have seen many references ...
- 63.9k