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

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71
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Is Fourier analysis a special case of representation theory or an analogue?

I'm asking this question because I've been told by some people that Fourier analysis is "just representation theory of $S^1$." I've been introduced to the idea that Fourier analysis is related to ...
56
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11answers
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Why aren't representations of monoids studied so much?

It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit ...
53
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3answers
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Does linearization of categories reflect isomorphism?

Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between ...
52
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7answers
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A learning roadmap for Representation Theory

As Akhil had great success with his question, I'm going to ask one in a similar vein. So representation theory has kind of an intimidating feel to it for an outsider. Say someone is familiar with ...
45
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3answers
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Is “semisimple” a dense condition among Lie algebras?

The "Motivation" section is a cute story, and may be skipped; the "Definitions" section establishes notation and background results; my question is in "My Question", and in brief in the title. Some ...
44
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9answers
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Is every finite group a group of “symmetries”?

I was trying to explain finite groups to a non-mathematician, and was falling back on the "they're like symmetries of polyhedra" line. Which made me realize that I didn't know if this was actually ...
43
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3answers
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5/8 bound in group theory

The odds of two random elements of a group commuting is the number of conjugacy classes of the group $$ \frac{ \{ (g,h): ghg^{-1}h^{-1} = 1 \} }{ |G|^2} = \frac{c(G)}{|G|}$$ If this number exceeds ...
43
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10answers
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Why are characters so well-behaved?

Last year I attended a first course in the representation theory of finite groups, where everything was over C. I was struck, and somewhat puzzled, by the inexplicable perfection of characters as a ...
43
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1answer
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Double affine Hecke algebras and mainstream mathematics

This is something of a followup to the question "Kapranov's analogies", where a connection between Cherednik's double affine Hecke algebras (DAHA's) and Geometric Langlands program was mentioned. I ...
42
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11answers
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Why is the exterior algebra so ubiquitous?

The exterior algebra of a vector space V seems to appear all over the place, such as in the definition of the cross product and determinant, the description of the Grassmannian as a variety, the ...
38
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20answers
8k views

Fun applications of representations of finite groups

Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not ...
38
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3answers
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What to do now that Lusztig's and James' conjectures have been shown to be false?

Lusztig and James provided conjectures for dimensions of simple modules (or decomposition numbers) for algebraic groups and symmetric groups in characteristic $p$. These conjectures have been ...
37
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5answers
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How much of the ATLAS of finite groups is independently checked and/or computer verified?

In a recent talk Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the ...
37
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2answers
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How much of character theory can be done without Schur's lemma or the Artin-Wedderburn theorem?

This is a somewhat imprecise question, as I am not sure how exactly how to formalise how to do mathematics "without" a certain key tool, but hopefully the intent of the question will still be clear. ...
37
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3answers
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Why is there such a close resemblance between the unitary representation theory of the Virasoro algebra and that of the Temperley-Lieb algebra?

For those who aren't familiar with the Virasoro or Temperley-Lieb algebras, I include some definitions: • The (universal envelopping algebra of the) Virasoro algebra is the $\star$-algebra $...
37
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2answers
4k views

Open problems/questions in representation theory and around?

What are open problems in representation theory? What are the sources (books/papers/sites) discussing this? Any kinds of problems/questions are welcome - big/small, vague/concrete. Some estimation ...
36
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1answer
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What is the status of Arthur's book?

Arthur's long-awaited book project is now published (The endoscopic classification of representations: orthogonal and symplectic groups). However, in the book he takes some things for granted: The ...
35
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1answer
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Consequences of Geometric Langlands

So, lots of people work on the Geometric Langlands Conjecture, and there have been a few questions around here on it (admittedly, several of them mine). So here's another one, tagged community wiki ...
35
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1answer
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Implications of non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials?

In their seminal 1979 paper here, Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the corresponding Iwahori-Hecke algebra. In particular they showed how to pass from a standard ...
34
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11answers
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What is significant about the half-sum of positive roots?

I apologize for the somewhat vague question: there may be multiple answers but I think this is phrased in such a way that precise answers are possible. Let $\mathfrak{g}$ be a semisimple Lie algebra (...
34
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7answers
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Why the Killing form?

I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...
34
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5answers
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Beautiful descriptions of exceptional groups

I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
34
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4answers
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What role does the “dual Coxeter number” play in Lie theory (and should it be called the “Kac number”)?

While trying to get some perspective on the extensive literature about highest weight modules for affine Lie algebras relative to "level" (work by Feigin, E. Frenkel, Gaitsgory, Kac, ....), I run into ...
34
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0answers
611 views

Class function counting solutions of equation in finite group: when is it a virtual character?

Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by $$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid w(x_1,\dots,...
33
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6answers
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How is representation theory used in modular/automorphic forms?

There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...
33
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7answers
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Lie group examples

I'm looking for interesting applications of Lie groups for an introductory Lie groups graduate course. In particular I'd like to hear of non-standard examples that at first sight do not seem to be ...
33
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2answers
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Why are parabolic subgroups called “parabolic subgroups”?

Over the years, I have heard two different proposed answers to this question. It has something to do with parabolic elements of $SL(2,\mathbb{R})$. This sounds plausible, but I haven't heard a ...
32
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4answers
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Intuition behind the definition of quantum groups

Being far from the field of quantum groups, I have nevertheless made in the past several (unsuccessful) attempts to understand their definition and basic properties. The goal of this post is to try to ...
32
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3answers
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Proper subgroup of GL(n,Z) isomorphic to GL(n,Z)?

This is just a question originated from some random thoughts. I hope it's nevertheless fit for mo. It's possible to find a proper subgroup of $GL(n,\mathbb{C})$ isomorphic to $GL(n,\mathbb{C})$ ...
32
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1answer
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Square roots of elements in a finite group and representation theory

Let $G$ be a finite group. In an an earlier question, Fedor asked whether the square root counting function $r_2:G\rightarrow \mathbb{N}$, which assigns to $g\in G$ the number of elements of $G$ that ...
31
votes
7answers
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Character table does not determine group Vs Tannaka duality

From the example $D_4$, $Q$, we see that the character table of a group doesn't determine the group up to isomorphism. On the other hand, Tannaka duality says that a group $G$ is determined by its ...
31
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5answers
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Definitions of Hecke algebras

There is a definition of Iwahori-Hecke algebras for Coxeter groups in terms of generators and relations and there is a definition of Hecke algebras involving functions on locally compact groups. Are ...
31
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4answers
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Induction and Coinduction of Representations

I'd like to understand the general framework of induction and coinduction of representations. If G is a finite group and H a subgroup, I know that there is a restriction functor from representations ...
31
votes
2answers
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Richness of the subgroup structure of p-groups

Given a prime $p$ and $n \in \mathbb{N}$, let $f_p(n)$ be the smallest number such that there is a group of order $p^{f_p(n)}$ which all groups of order $p^n$ embed into. What is the asymptotic growth ...
30
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4answers
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Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?

A $4\times 4$ symmetric matrix $$ \left( \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12} & a_{22} & a_{23} & a_{24} \\ a_{13} & a_{23} & a_{33} & ...
30
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8answers
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Ubiquity of the push-pull formula

The push-pull formula appears in several different incarnations. There are, at least, the following: 1) If $f \colon X \to Y$ is a continous map, then for sheaves $\mathcal{F}$ on $X$ and $\mathcal{G}...
30
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4answers
1k views

Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?

Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are ...
30
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2answers
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What do cluster algebras tell us about Grassmannians?

One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
30
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1answer
912 views

Roadmap to Geometric Representation Theory (leading to Langlands)?

I believe there has been at least one question similar to this one and yet I still think this particular question deserves to have a thread of its own. I'm becoming increasingly fascinated by stuff ...
29
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2answers
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Are there “real” vs. “quaternionic” conjugacy classes in finite groups?

The complex irreps of a finite group come in three types: self-dual by a symmetric form, self-dual by a symplectic form, and not self-dual at all. In the first two cases, the character is real-valued, ...
28
votes
10answers
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The finite subgroups of SL(2,C)

Books can be written about the finite subgroups of $\mathrm{SL}(2,\mathbb C)$ (and their immediate family, like the polyhedral groups...) I am about to start writing notes for a short course about ...
28
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6answers
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What is known about this plethysm?

Let $S^{\lambda}$ be a Schur functor. Is there a known positive rule to compute the decomposition of $S^{\lambda}(\bigwedge^2 \mathbb{C}^n)$ into $GL_n(\mathbb{C})$ irreps? In response to Vladimir'...
28
votes
2answers
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Can an operator have Exp(z) as its characteristic “polynomial”?

Let $\mathcal{H}$ be a Hilbert space, and let $T: \mathcal{H} \rightarrow \mathcal{H}$ be a trace-class operator. Define $$ f_T(z) = \sum_{i=0}^\infty \mbox{Tr}(\wedge^k T) \cdot z^k, $$ the ...
28
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2answers
675 views

How to make the Capelli's identity less mysterious?

The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity To ...
28
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3answers
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Being a subgroup: proof by character theory

Let me first cite a theorem due to Frobenius: Let $G$ be a finite group, with $H$ a proper subgroup ($H\ne (1)$ and $G$). Suppose that for every $g\not\in H$, we have $H\cap gHg^{-1}=(1)$. Then ...
27
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17answers
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ADE type Dynkin diagrams

The ADE type Dynkin diagrams seem to come up in seemingly different areas of math. Two places they come up are: (1) Classification of simply laced complex simple lie algebras. (2) Finite subgroups ...
27
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5answers
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Deformation theory of representations of an algebraic group

For an algebraic group G and a representation V, I think it's a standard result (but I don't have a reference) that the obstruction to deforming V as a representation of G is an element of H2(G,V&...
27
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9answers
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How is the physical meaning of an irreducible representation justified?

This is maybe not an entirely mathematical question, but consider it a pedagogical question about representation theory if you want to avoid physics-y questions on MO. I've been reading Singer's ...
27
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5answers
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Does $S_4$ inject into $SL(2,R)$ for some commutative ring $R$?

$\newcommand{\Z}{\mathbf{Z}}$ Given a nice infinite collection of groups, for example the symmetric groups, one can ask whether any finite group is a subgroup of one of them. Of course any finite ...
27
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0answers
874 views

Why are there so few quaternionic representations of simple groups?

Having spent many hours looking through the Atlas of Finite Simple Groups while in Grad school, I recall being rather intrigued by the fact that among the sporadic groups, only one (McLaughlin as I ...