Questions tagged [rt.representation-theory]
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
7,076 questions
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Where do the real analytic Eisenstein series live?
In obtaining the spectral decomposition of $L^2(\Gamma \backslash G)$ where $G=SL_2(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup (I am satisfied with $\Gamma = SL (2,\mathbb{Z})$) we have a ...
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Typos in Bourbaki's root-system tables
A while ago, a colleague told me that he thought he remembered that there were typos in Bourbaki's tables in the English translation of "Groupes et algèbres de Lie", but that he could no ...
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The sum (with multiplicity) of the cubes of irreducible character degrees of a finite group
Throughout $G$ is a finite, non-abelian group.
$\DeclareMathOperator\Irr{Irr}\DeclareMathOperator\AD{AD}\DeclareMathOperator\cp{cp}\newcommand\card[1]{\lvert#1\rvert}$
Let $\Irr(G)$ be the set of ...
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Gap in an argument in Fulton & Harris?
I'm reading through the two chapters in Fulton and Harris on the representation theory of $\mathfrak{sl}(3,\mathbb{C})$, in preparation for lecturing on them this week. I'll use F&H's notation, ...
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Variety of commuting matrices
Let $G=\operatorname{GL}(n,\mathbb{C})$ and $\mathfrak{g}=\operatorname{Mat}(n,\mathbb{C})$ and let us consider the two varieties $X,Y$ defined as $$X=\{(x,y) \in G \times G \ | \ xy=yx\} $$ and $$Y=\{...
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$\text{Rep}(D(G))$ as representation category of a vertex operator algebra
The category of representations $\text{Rep}(D(G))$ of the quantum double of a finite group is well-known to be a modular tensor category. Can these modular tensor categories also be obtained as ...
- 2,513
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References for Langlands classification
I kindly ask about some references concerning the representation theory of the Langlands dual of a compact Lie group, and how it relates to things related to the original compact Lie group.
My ...
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Quantum cohomology of partial flag manifolds
Is there a place in the literature where the quantum differential
equation (or even just quantum cohomology algebra)
of partial flag manifolds $G/P$ is computed for arbitrary semi-simple $G$ and
...
- 7,037
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How complicated is infinite-dimensional "undergraduate linear algebra"?
The name "undergraduate linear algebra" in the title is a bit of a joke, and so I don't know how widely spread it is. To wit:
High school linear algebra is the theory of a finite-dimensional vector ...
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Character of parity-twisted supersymmetric VOA module -- question inspired by the Stolz-Teichner program
I'll begin with some background that is unnecessary for the actual question, but that might be interesting to the reader:
Topological modular forms ($TMF$) is a generalized cohomology theory whose ...
- 43.2k
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For which finite groups $G$ is every character a virtual permutation character?
Let $G$ be a finite group. A (complex) character $\chi$ of $G$ is said to be a virtual permutation character if it can be expressed as a $\mathbb{Z}$-linear combination of characters induced from the ...
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Which L-functions are not "Langlands-Shahidi L-functions"?
The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...
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Equivariant Möbius inversion
I'll first explain what Möbius inversion says, and then state what I am fairly sure the equivariant version is. I can write out a proof, but I also can't believe this hasn't been done already; this is ...
- 156k
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Why do these two Monster-related calculations yield $163$?
Fact 1: (1979, Conway and Norton)$^{1}$
"There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster."
Note: There are 194 (linear) irreducible ...
- 12.6k
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Examples of representations of quantum groups
I am trying to learn some basic theory of quantum groups $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a simple Lie algebra, say $sl_n(\mathbb{C})$. As far as I heard the finite dimensional ...
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D-modules over algebraic curves VS differential Galois theory
Disclaimer: I know very little about both of the fields in question.
My question is pretty simple:
What's the relation between differential Galois theory and D-modules
over algebraic curves?
...
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Freeness of tensor product
Let $G$ be a finite group. Is $\mathbb{Z}G\otimes_{Z(\mathbb{Z}G)}\mathbb{Z}G$ free as a $\mathbb{Z}$-module, where $Z$ denotes the centre?
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What is the recent development of D-module and representation theory of Kac-Moody algebra?
I just started to collect the papers of this field and know little things. So if I make stupid mistake, please correct me.
It seems that there are several approaches to localize Kac-Moody algebra(in ...
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Noether's bound for anticommutative invariant theory (diff. forms instead of polynomials)?
EDIT: Now with a concrete request to CAS experts (see the end of the post).
Let $G$ be a finite group, and $V$ a finite-dimensional representation of $G$. The classical invariant theory of $G$ and $V$...
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Reconstruction Conjecture holds for Directed Acyclic Graphs?
Wikipedia's article on the Reconstruction Conjecture mentions that the conjecture is false for digraphs, and refers to two papers by Stockmeyer. As far as I can see, none of the counter-examples in ...
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Number of $F_p$-matrices ac=ca, bd = db , ad - da = cb - bc is polynomial in p ? ("Manin matrix variety" - normal ? Cohen–Macaulay ? )
Consider four $n\times n$ matrices $a,b,c,d$ over finite field $F_q$ (or $F_p$ for simplicity), such that they satisfy three equations: $ac=ca,bd=db, ad-da=cb-bc $. Thus an affine algebraic manifold ...
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Symmetries of local systems on the punctured sphere
Let $X=S^2\setminus D$, for $D\subset S^2$ some finite set of points, say with $|D|=n\geq 1$. The category of locally constant sheaves of $\mathbb{C}$-vector spaces on $X$ (equivalently, complex ...
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When is the determinant an $8$-th power?
I am working over $\mathbb{R}$ (though most of the story goes over any field). I am looking for linear spaces of matrices such that the restriction of the determinant to this spaces can be written (...
- 7,300
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Monstrous Langlands-McKay or what is bijection between conjugacy classes and irreducible representation for sporadic simple groups?
Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group and some other groups there is "good ...
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What to do with results you found but cannot prove(outside your research area)?
Not sure if MathOverflow is still a place to discuss such things, but I'll give it a try. Tell me an alternative site, in case it is wrong here. I translated a representation-theory/combinatorial ...
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Does a symplectic group act on a tensor power of a spin representation?
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}$More specifically, let $S_k$ be the spin representation of $\Spin(2k+1)$.
Then is there are action of $\Sp(2r-2)$ on $\bigotimes^{2r}S_k$ ...
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Combinatorial identity involving the Coxeter numbers of root systems
The setup is:
$R$ = irreducible (reduced) root system;
$D$ = connected Dynkin diagram of $R$, with nodes numbered $1,2,...,r$;
$\hat D$ = extended Dynkin diagram, nodes numbered $0,1,2,...,r$;
$\...
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Integration of a function over 7-sphere
Suppose we have $x_1^2 + y_1^2 + x_2^2 + y_2^2 + x_3^2 + y_3^2 + x_4^2 + y_4^2 = 1$ and we define $z_j = x_j + iy_j$, where $j = 1,\,2,\,3,\,4$.
The problem is finding or approximating the ...
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Is there a connection between representation theory and PDEs?
As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination ...
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Decompose tensor product of type $G_2$ Lie algebras.
Let $G$ be a semisimple Lie algebra over $\mathbb{C}$. Let $V(\lambda)$ be the irreducible highest weight module for $G$ with highest weight $\lambda$. If $G$ is of type A, we can decompose $V(\lambda)...
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Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$
$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power $V^...
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Generators of invariant polynomials of semisimple Lie algebra
Suppose $\mathfrak{g}$ is a complex semi-simple Lie algebra. By a theorem of Chevalley, we know that $S(\mathfrak{g})^\mathfrak{g}$, i.e. the $\mathfrak{g}$ invariant polynomials, is generated by $l$ ...
- 587
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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 &...
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HOMFLY and homology; also superalgebras
My understanding is that an analogy along the following lines is (roughly) true:
"The Alexander polynomial is to knot Floer homology is to gl(1|1)
as the Jones polynomial is to Khovanov homology is ...
- 2,806
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Is the cohomology ring $H^*(BG,\mathbb{Z})$ generated by Euler classes?
I am interested in the classifying space $BG$ of a finite group $G$.
A real representation $V$ of $G$ of dimension $r$ defines a real vector bundle over $BG$ of rank $r$. If the determinant of this ...
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How to compute all irreducible representations of a finite group ? (how GAP is doing this?)
Let us "take" a finite group G. Here "take" I mean any type of group-theoretic description you prefer: e.g. as an explicit subset of GL (or other group) or Cayley table, whatever.
Question: How ...
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Conjectures in the representation theory of the symmetric group
Question: What are current open conjectures about the representation theory of the symmetric group?
I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
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Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras
Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$-module divides the dimension of $H$.
In which cases the conjecture is known ...
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Maximal number of maximal subgroups
Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...
- 2,179
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Etymology of cuspidal representations
In the literature on representation theory of $GL_2(\Bbb F_p)$ and $GL_2(\Bbb Q_p)$, the irreducible representations with trivial Jacquet module are often called "cuspidal" or "supercuspidal". Why are ...
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An algebraic approach to the thermodynamic limit $N\rightarrow\infty$?
In physics one studies quite often the thermodynamic limit or what we call the $N\rightarrow \infty$ behavior of a system of $N\rightarrow\infty$ particles. This is of particular relevance in the ...
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Making the branching rule for the symmetric group concrete
This question concerns the characteristic $0$ representation theory of the symmetric group $S_n$. I'm a topologist, not a representation theorist, so I apologize if I state it in an odd way.
First, a ...
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On Category O in positive characteristic
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$. In the case that $k$ has characteristic 0, there has been intensive study of the BGG category O of representations of ...
- 3,637
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Is there a relationship between Broué's abelian defect group conjecture and Alperin's weight conjecture?
Let $G$ be a finite group, let $k$ be a large enough field of characteristic $p>0$. Let $p\mid |G|$.
Broué's abelian defect group conjecture states the following:
Let $B$ be a block of $kG$ with
...
- 1,755
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How complicated can a finite double complex over a field be?
A finite complex over a field $k$ is pretty simple: it's the direct sum of its homology with a split-exact complex. How complicated can a finite double complex be? Does it make a difference if $k$ is ...
- 63.9k
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A careful roadtrip from locally symmetric spaces to algebra
I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning completely)...
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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 ...
- 390
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How do I know if an irreducible representation is a permutation representation?
I have a vague question, a less vague question and a lot of vaguer questions about permutation representations of a finite group $G$.
Vague question. Recall that if $G$ acts on a finite set $X$, we ...
- 575
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Tensor power of the natural representation of Sn
The symmetric group $S_n$ acts over $V=\mathbb{R}^n$ by permuting the canonical basis.
So it acts over $V^{\otimes p}$ with a diagonal action (acts the same over each element of the tensor product).
...
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