Tagged Questions
1
vote
0answers
19 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
63 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.
...
11
votes
1answer
91 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
36 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 ...
8
votes
0answers
58 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
52 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
32 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 $\#G$ ...
6
votes
2answers
104 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
28 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
45 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
47 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 ...
4
votes
2answers
77 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
51 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
27 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
17 views
References request: Introduction to K3 surface.
Are there some good books or survey papers about K3 surface? Thank you very much.
6
votes
2answers
103 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 ...
3
votes
1answer
92 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
52 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?
6
votes
0answers
70 views
The center of a simply connected semisimple Lie group
I am learning about Lie groups, and I have the following basic question:
Every Lie group $G$ has a (unique) universal covering group $ \bar G $ that is simply connected, and such that the covering ...
2
votes
1answer
101 views
Conjugate Representations
Are there any general results on when conjugate representations of a real Lie algebra are equivalent? I'm inclined to say that they are often not, but this is merely going on my case by case ...
8
votes
1answer
277 views
Undergraduate roadmap for Langlands program and its geometric counterpart
What are the topics which an undergraduate with knowledge of algebra, galois theory and analysis learn to understand Langlands program and its goemetric counterpart? I would also like to know what are ...
3
votes
0answers
52 views
Induced representation and degrees
Let $G$ be a finite group, $S$ a subgroup, $K$ an arbitrary (not necessarily algebraically closed) field which characteristic does not divide the group order.
1) Let $Ind_S^G(f)$ have the degree $n$ ...
6
votes
1answer
64 views
What theorems/examples will make me really understand representation theory?
Okay, so I've been through some basic results on representation theory. I've gone over the proof of Burnside's $pq$ theorem using characters. I've also read though the basics of Lie groups and ...
6
votes
1answer
74 views
Formula for evaluation of character on a transposition
Let $\lambda\vdash n$ be a partition of $n\in\mathbb N$ and $\chi=\chi_\lambda$ the corresponding irreducible character of the symmetric group $S_n$. Denote by $\lambda^t$ be the transpose of ...
4
votes
2answers
88 views
Classification of irreducible representations via Casimirs
Physicists almost always label irreducible representations via Casimirs (e.g., characterizing the irreducible representations of $SO(3)$ by spin). I've been looking far and wide to see the general ...
1
vote
2answers
105 views
Is there formula name and proof for this theorem ? ( guess it's called Burnside character formula)
The formula answers: how many tuples $(\sigma_1,\sigma_2,\dots,\sigma_n)$ of elements of a given group $G$ such that
(1) $\sigma_i\in C_i$ , where $C_i$ stands for conjugacy class.
(2) ...
4
votes
2answers
208 views
$SU(2)$ Representation of $SO(3)$
I've often seen it written that $SU(2)$ is a "two-valued representation" of $SO(3)$ (in theoretical physics books mainly). I have a major conceptual issue with this however.
I know there is a Lie ...
4
votes
2answers
136 views
Looking for texts in representation theory
I recently finished a course in representation theory, and while I learned a lot from it, I know that there's a lot more in the subject that I missed. For the course we used Fulton and Harris as a ...
3
votes
2answers
127 views
Schur-Weyl Duality ( Classical ) and the Double Commutant reference request
I would like to ask for any reference suggestions on the topic of Schur-Weyl Duality for GLn ( directly GLn, not through the lie algebra ) and the double commutant theorem.
The section on this ...
5
votes
1answer
66 views
Dimension of the GL-orbit of d-forms in one less variable
Let $V:=k[x_0,\ldots,x_n]_d$ be the $k$-vector space of homogeneous polynomials of degree $d$. Let $G:=\mathrm{Gl}(n+1,k)$ act on $V$ induced by the canonical action on the linear forms: For ...
2
votes
1answer
67 views
Sum of squares of dimensions of irreducible characters.
For anyone familiar with Artin's Algebra book, I just worked through the proof of the following theorem, which can be seen here:
(5.9) Theorem Let $G$ be a group of order $N$, let ...
2
votes
1answer
97 views
Trace and identity are the only linear matrix invariants?
This question is obviously related to that recent question of mine, but I feel it’s sufficiently different to be posted as a separate question. Let $V$ be a finite-dimensional space. Let ${\cal L}(V)$ ...
1
vote
2answers
128 views
Number of prime divisors of element orders from character table.
From wikipedia:
It follows, using some results of Richard Brauer from modular representation theory, that the prime divisors of the orders of the elements of each conjugacy class of a finite group ...
7
votes
0answers
75 views
Reference for l-adic Lie algebras
I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'.
Is there a standard reference for ...
0
votes
0answers
27 views
Irreducible $2$-dimensional $\mathbb{C}$-representation of $D_{16}\times C_{2}$.
I am looking for all irreducible $2$-dimensional $\mathbb{C}$-representation of $D_{16}\times C_{2}$, where $D_{16}$ is the dihedral group of order $16=8\times 2$ and $C_{2}$ is the cyclic group of ...
1
vote
0answers
27 views
Smallest dimensional irreps of semi-simple Lie algebras
I'm wondering if there is a reference that lists the first couple smallest dimensional irreducible representations of each semi-simple Lie algebra. I know these can be found using the Weyl dimension ...
5
votes
1answer
150 views
Methods of Multilinear Algebra in Representation Theory
I have been interested in representation theory lately in particular on that of Lie algebras. Now I have noticed that one way of building representations is to take tensor/exterior/symmetric powers. I ...
4
votes
2answers
83 views
is there a better classification of a desirable algebra?
Consider a finite-dimensional, associative algebra presented as follows:
$$\mathcal{A} = e_1 \mathbb{R}\oplus e_2 \mathbb{R} \oplus \cdots \oplus e_n \mathbb{R} $$
with multiplication $*: \mathcal{A} ...
17
votes
9answers
795 views
What's a good place to learn Lie groups?
Ok so I read the following article the other day: http://www.aimath.org/E8/ and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought "oh it must have something to ...
4
votes
4answers
578 views
Best books on Representation theory
What are some of the best books on Representation theory for a beginner? I would prefer a book which gives motivation behind definitions and theory.
3
votes
3answers
70 views
How to expand a representation
If there is a finite group with a normal subgroup and a representation of this subgroup over a finite field. How can one expand this representation to a representation of the whole group? Are there ...
3
votes
0answers
66 views
Semidirect product of reductive groups
Given two linearly reductive algebraic groups, is their semidirect product reductive again? By linearly reductive, I mean that any rational representation of the group is completely reducible. In ...
3
votes
2answers
200 views
Representation of Cyclic Group over Finite Field
The post Irreducible representations of a cyclic group over a field of prime order discusses the irreducible representations of a cyclic group of order $N$ over a finite field $\mathbb{F}_p$ where $N$ ...
4
votes
0answers
67 views
Law of large numbers for Plancherel random Young diagrams
Do you know a reference book on the law of large numbers for random Plancherel Young diagrams ? I know the book of Kerov, but actually, it is only a compilation of his articles, and i need something ...
2
votes
0answers
39 views
Bijection between $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$ and lattices in $F^n$
I've come across mention of a bijection between lattices in $F^n$ ($F$ a field, in my case $\mathbb{C}(\!(t)\!)$) and elements of $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$, where $O$ is the ring ...
3
votes
1answer
92 views
The Frobenius-Nakayama Formula
I am currently reading a paper where it refers to the usual Frobenius-Nakayama formula describing quotients of an induced module. It is refering to the following result:
If $k$ is a field, $P$ is ...
2
votes
0answers
64 views
Classifying continuous characters $\epsilon:\mathbf{C}^\times\to \mathbf{C}^\times$.
I recently saw the following claim: Let $\mathbf{C}$ denote the field of complex numbers together with its usual topology. If $\epsilon:\mathbf{C}^\times\to \mathbf{C}^\times$ is a continuous ...
2
votes
0answers
38 views
Finite-dimensional representations of the Lie algebra of vector fields on a circle
I have just began to study infinite-dimensional Lie algebras and I am curious whether the Lie algebra $L$ spanned by the vector fields $z^n \partial/\partial z$, $n=0,1,2,3,\dots$ admits any ...
0
votes
0answers
55 views
Representations of Central Products
What is a good reference for learning about representations/characters of central products of groups?
By central product, I mean the following.
If $G$ and $H$ are groups, containing isomorphic ...
1
vote
0answers
42 views
How to characterize the image of $PGL(V)$ in $\mathbb{P}(W)$ for an irreducible $GL(V)$-representation $W$
I'd like to ask two versions of my question, one a very specific case that I suspect may have a fairly easy and classical answer, the other the general case which I would like to find references on.
...
