2
votes
1answer
62 views

The comultiplication on $\mathbb{C} S_3$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=S_3$, ...
9
votes
2answers
206 views

History of the matrix representation of complex numbers

It is well-known to many that $\mathbb{C}$ can be represented by matrices of the form $\left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$. For example, see this question or this ...
1
vote
0answers
46 views

Positive definite functions generated by irreducible representations — what do people call them?

Let $G$ be a group and $\pi:G\to B(H)$ be its irreducible unitary representation (one can endow $G$ with topology and claim that $\pi$ is continuous in some sense, this doesn't matter). For a given ...
10
votes
2answers
172 views

Applications of Algebra in Physics

Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics. I have a good preparation in Algebra and ...
1
vote
1answer
43 views

Finding a lie group structure on $\mathbb R^n\setminus\{0\}$

I want to find all maps $g: \mathbb R^n\setminus \{0\} \rightarrow GL_n(\mathbb R)$ which satisfy the properties $g$ is differentiable and injective $g(g(a)b) = g(a)g(b)$ for all $a,b\in\mathbb ...
2
votes
0answers
25 views

$\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules

Let $p$ be prime. Let $\mathbb{Z}_p$ be the $p$-adic integers. I'm intersted in $\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules of low rank over $\mathbb{Z}_p$ (rank $2$ is already very ...
0
votes
0answers
19 views

Ordinary irreducible representations of semi-direct products

What is the best source for learning about constructing irreducible representations of semi-direct product $G=N \rtimes_\phi H$ from irreducible representations of $H$ and $N$ over field of complex ...
2
votes
1answer
82 views

Quasi-hereditary algebras

Can anyone please recommend a reference (I prefer a book chapter) on Quasi-hereditary algebras? and if it is possible tell me how to prove that Schur algebras are Quasi-hereditary (or any other ...
0
votes
1answer
79 views

The number of a set of irreducible projective characters vs the number of the ordinary characters of a finite group G.

I need valid references to show that the number of a set of irreducible projective characters with non-trivial factor set is always strictly less than the number of the ordinary characters of a ...
5
votes
2answers
75 views

Reference Request: Characters of Finite General Linear Groups

I've been looking at J.A. Green's article The Characters of Finite General Linear Groups and it seems that Green in this article comes up with a way of calculating all irreducible characters of a ...
5
votes
1answer
111 views

Representation theory over $\mathbb{Q}$

I am looking for books or papers which tell me something about representation theory of finite groups over $\mathbb{Q}$ (or finite extensions thereof which are not splitting fields of the group ...
2
votes
1answer
43 views

Why does the universal cover of $GL^+_n$ not admit finite-dimensional representations?

Let $GL^+_n \subset \mathbb{R}^{n \times n}$ be the subgroup of real matrices with positive determinant and $\widetilde{GL}^+_n$ be its universal cover. Why does $\widetilde{GL}^+_n$ not admit ...
1
vote
1answer
64 views

Reference for Weyl Character Formula

I am reading Lie algebra book by James E.Humphreys. This book giving enough discussion about Weyl Character Formula and its proof, still I would like to know what are the other books or lecture notes ...
5
votes
2answers
91 views

History of representation theory

For a student's journal I want to write a short article about history and importance (applications) of representation theory. Are there some accesible literature about this?
1
vote
1answer
28 views

representations into $\mathfrak{sl}(n,\mathbb{C})$

in Borel/Ji "Compactification of symmetric and locally symmetric spaces" the standard Satake compactification is constructed and general Satake compactifications are realized via an embedding into the ...
2
votes
1answer
63 views

What is $\mathfrak{gl}(\infty)$

As title says, I know what is $\mathfrak{gl}(n,\mathbb{C})$, but what is $\mathfrak{gl}(\infty)$? Where can I find good reference for this?
6
votes
2answers
91 views

Eigenbundle decomposition

Let $G$ be a finite cyclic group and $X$ a smooth manifold equipped with a trivial $G$-action. It is known that we can decompose every $G$-equivariant vector bundle with respect to the action: ...
1
vote
4answers
181 views

Representation Theory book other than Fulton's

Fulton/Harris's book on representation theory seems to be the "definitive" introductory text on the subject. But is there perhaps a lower level introduction to the subject? Most of the very first ...
2
votes
1answer
151 views

Finding Idempotents?

I was wondering if there is a method for finding primitive idempotents of a finite dimensional algebra (over a field)? or in other words is there any way to build the complete set of primitive ...
3
votes
2answers
148 views

Reference request for studying Lie group & Lie algebra representations

I am learning representation theory of Lie groups & Lie algebras from the book by Brian Hall. Unfortunately, this does not discuss infinite dimensional representations. Which books should I study ...
4
votes
1answer
59 views

reference request: proof that group characters are a basis for $L^2$

I know the following must be very standard, but I haven't found it in any of the functional analysis books to which I have access. Do you know where I can find a self-contained proof?: If $G$ is a ...
3
votes
1answer
92 views

Research paper in harmonic analysis that can be read in parallel to studying the subject.

In my idle hours I started to learn some math I touched only superficially in academia in former times. Among others I am working through the books of A. Deitmar on harmonic analysis. I've almost ...
4
votes
1answer
129 views

Clifford Algebra for understanding Atiyah Singer Index Theorem Reference Request

I am interested in studying Atiyah Singer Index Theorem and Spin Geometry and would like to study Clifford Algebras and their representations for this purpose. I have a book 'Clifford Algebras : An ...
4
votes
1answer
85 views

Proof that ideal of Plücker relations is a prime ideal

I am reading section 8.4 of Fulton's Young tableaux where he defines a certain ring as follows. Fix a complex vector space $E$ of dimension $m$ and integers $d_1,\ldots d_s$ such that $m \geq d_1 > ...
1
vote
1answer
59 views

Reference request for Modular representation theory

I am trying to learn modular representation theory. I would be thankful if any one tell me a good reference to start with?
3
votes
1answer
69 views

Representation of topological groups

I am looking for a good book of topological representation. I have a very good insight of representation theory of finite groups, and I want to explore topological representations. I saw a book by ...
6
votes
0answers
131 views

The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group

Question What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group? The trivial corepresentation is given by $\Delta_{|W}$ where $W$ is just the one dimensional ...
3
votes
1answer
39 views

Number of indecomosbale $\mathbb{Z}_p[G]$ modules finite

Is there a theorem like those of Jones, which tells if the number of different $\mathbb{Z}_p[G]$ modules is finite, where $G$ is a finite group and $\mathbb{Z}_l$ the $p$-adic ring?
5
votes
1answer
80 views

Very basic question on continuous representations

Suppose $G$ is a topological group, i.e. $$G \times G \to G, (g,h) \mapsto gh ~~~~ G \to G, g \mapsto g^{-1}$$ are continuous. Suppose $K$ is a field with norm $|\cdot|$ on it and $V$ is a normed ...
8
votes
1answer
426 views

Applications of representation theory in physics

The notes of a lecture on basic group and representation theory I attended last semester begin with a bit of motivation for the argument. They give the following examples for applications in physics: ...
8
votes
1answer
331 views

Learning Roadmap for Borel - Weil - Bott Theorem

Next semester I may study a course where the ultimate goal is to get to the Borel - Weil - Bott (BWB) Theorem, if not at least try to understand it in the case that we have $G = \text{SL}_n$. I have ...
6
votes
1answer
437 views

Online course in representation theory or differential geometry

Are there any courses in representation theory that are available online? I'm looking for a course including videos, notes as well as assignments. I'd also be interested in a course in differential ...
1
vote
0answers
25 views

How to write down the maximal subgroups of $GL(9, \mathbb{C})$

I am wondering about the maximal subgroups of the group $GL(n^2, \mathbb{C})$. My motivation for wondering about these groups is a project (in its most general form) I am working on where I am trying ...
3
votes
3answers
147 views

Book recommendation for associative algebras

Currently, I am reading David Radford's Hopf Algebra, and I would like to pick up some representation theory of associative algebras as well since my knowledge of them is pretty shallow at the moment. ...
12
votes
1answer
191 views

Introduction to the trace formula for people outside number theory

I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula. I am aware that there are "standard references" on the ...
5
votes
1answer
48 views

Connection between $\mathbb{Q}_p[G]$ and $\mathbb{Z}_p[G]$

In this post there was the comment, that having $\mathbb{Q}_p[G]$ modules, it is possible to construct $\mathbb{Z}_p[G]$ modules. How is it possible to find out when there is a bijection between ...
6
votes
0answers
116 views

Expression of basis vectors of permutation modules in different bases.

Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that corresponds to $\lambda$, i.e. the complex vector space spanned by all standard ...
3
votes
2answers
135 views

Frobenius reciprocity

I would like to ask a question on Theorem 8.6 on page 246 in this book. There is the claim that the multiplicity of $F$ in $E^G$ is equal to the multiplicity of $E$ in $F_H$. Why is this just ...
2
votes
0answers
51 views

Why does $d_{\alpha}$ divide $\#G$ for $\alpha\in\hat{G}$?

Let $\alpha$ be a unitary irreducible representation of a finite group $G$. Then we have \begin{equation} d_{\alpha}|\#G, \end{equation} where $d_\alpha$ is the degree of the representation and $\#G$ ...
7
votes
2answers
489 views

Why are (representations of ) quivers such a big deal?

Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, ...
3
votes
1answer
70 views

Induced representation is isotypical?

Is there a theorem like this for the induced representation? Let $N$ be a normal subgroup of a finite group $G$ and $\rho$ be an irreducible linear representation over any field $k$. Then one of ...
2
votes
0answers
54 views

Subspaces stabilized by representations of $\mathrm O(9)$

I am trying to figure out what representations of maximal subgroups of $\mathrm{GL}_{n^2}$ stabilize one dimensional subspaces in $\mathrm{GL}(\mathrm{Sym}^n(\Bbb C))$. More precisely, let the setup ...
0
votes
1answer
70 views

Representations over $\mathbb{Q}_p$

I would like to understand representations over the $p$-adic field $\mathbb{Q}_p$ and find simple $\mathbb{Q}_p[G]$ modules for a finite group $G$. Is there some famous literature like for ...
5
votes
2answers
232 views

Good book on representation theory after reading Rotman

I'm about to finish Rotman's "Introduction to the Theory of Groups" and I would like to continue my study of group theory with a book on representation theory. The book should give a broad overview ...
0
votes
1answer
120 views

Topic for presentation on Group Representations, Young Tableaux, Symmetric Group

I need to do a presentation relating to group representations/Young tableaux/symmetric group; however, for all my searching, I cannot find a cool topic that I find personally interesting (and that is ...
1
vote
1answer
126 views

Definition of induced representation by tensor product

Suppose there is a finite group $G$ with a subgroup $H$ an some field $K$. If one has a representation of the subgroup, one can construct the induced representation $\rho:=Ind_H^G$ according to ...
0
votes
1answer
52 views

References request: Introduction to K3 surface.

Are there some good books or survey papers about K3 surface? Thank you very much.
7
votes
2answers
156 views

Do group rings appear outside of representation theory?

I am particularly concerned with finite groups. I have seen group rings used in the fundamentals of representation theory as the dual notion to representations. I haven't ever seen them anywhere ...
4
votes
1answer
211 views

Irreducible subgroups of $\text{GL}(2,p)$

I just wonder any sources that provides the list of all minimal irreducible subgroups of $\text{GL}(2,p)$. Here the term "minimal" means there is no proper subgroup that is irreducible. A list of ...
5
votes
1answer
123 views

A proof of the Weyl Character formula via fixed point formula and

I've been looking all day for a reference or notes that prove the Weyl character formula via a fixed point formula and the Borel-Weil-Bott theorem. Does anyone know of these off hand?