Tagged Questions

Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.

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-1
votes
2answers
30 views

Representation Theory. Why does $a^{r}b^{s}a^{t}b^{u} = a^{i}b^{j}$?

I want to know that why we have $a^{r}b^{s}a^{t}b^{u} = a^{i}b^{j}$. Please let me know.
0
votes
0answers
19 views

Symmetry of Plancherel measure (for $S_n$)

For each $n \geq 1$ consider the reverse lexicographical order on the set $P(n)$ of partitions of $n$. Example for $n=7$: $$ \begin{pmatrix} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 ...
1
vote
0answers
14 views

A technical problem on constructing smooth vectors in a representation

Let $G$ be a linear Lie group, say $\mathrm{SL}(2,\mathbb{R})$, and $\pi:G\to\mathrm{GL}(H)$ a continuous representation of $G$ in a Banach space $H$. Let $\pi^1:\mathcal{C}_c(G)\to\mathrm{End}(H)$ be ...
0
votes
0answers
15 views

Representation of even/odd type

I have seen in my reading in Dynkin's Maximal Subgroups of the Classical Groups that an irreducible representation $\phi$ of $A_n, D_{2k+1},$ or $E_6$ has a bilinear invariant if and only if the ...
6
votes
1answer
96 views
+250

Decompose $P$ into the direct sum of irreducible representations.

Note: I need help with part (c). Consider the representation $P: S_3 \rightarrow GL_3$ where $P_{\sigma}$ is the permutation matrix associated to $\sigma$. a) Determine the character $\chi_P : S_3 ...
0
votes
1answer
22 views

Complex representation and Dual representation notation

Let's say we have a representation $\rho$ of $G$ on a vector space $V$. Wikipedia refers to the dual representation as $V^*$, but the dual vector space as $\overline{V}$. It does the opposite for the ...
4
votes
1answer
76 views

How to understand $\frac{d}{dt}\{(\exp(tX))_*(Y)\}|_{t=0}=[X,Y]$?

Let $G$ be a Lie group on which $X$ and $Y$ are two vector fields. Let $G\xrightarrow{\exp(tX)} G$ be the (Lie theory) exponential map corresponding to $X$. Then of fundamental importance is ...
4
votes
1answer
53 views

Why is Lie derivative smooth?

Let $G$ be a linear Lie group, say $\mathrm{SL}(2,\mathbb{R})$. Suppose $X\in\mathfrak{g}$ and $f:G\to\mathbb{R}$ is smooth. The Lie derivative of $f$ with respect to $X$ is the function ...
6
votes
0answers
191 views

Plancherel formula for compact groups from Peter-Weyl Theorem

I'm trying to derive the following Plancherel formula: $$\|f\|^{2}=\sum_{\xi\in\widehat{G}}{\dim(V_{\xi})\|\widehat{f}(\xi)\|^{2}}$$ from the statement of the Peter-Weyl Theorem as given by Terence ...
1
vote
1answer
19 views

Invariant hermitian forms and irreducible representations

Let $V$ be a vector space over $\mathbb{C}$ of finite dimension $n$, $G$ is a finite group and $T:G\rightarrow GL(V)$ its irreducible representation that sends each $g$ into $T_g$. Let $E:V^{\bigoplus ...
4
votes
1answer
43 views

Group homomorphisms into a field

Let $G$ be a finite group, and let $k$ be a field, which should be algebraically closed, I think. How to describe all homomorphisms $G\rightarrow k^*$ (i.e. one-dimensional representations: ...
7
votes
1answer
41 views

Specific projective dimension of a module over bound quiver

Suppose $K$ is an algebraically closed field, and $A$ is the algebra presented by the quiver $$\require{AMScd} \begin{CD} 1 @>>> 2\\ @V{}VV @V{}VV \\ 3 @>>> 4 @>>> 5 ...
2
votes
1answer
42 views

Does an irreducible $\mathbb CG$-module have a basis of the form $u,ug_1,\dots,ug_n$?

Suppose that $U$ is an irreducible $\mathbb CG$-module and $u\in U$. Let $\operatorname{span}(u_1,\dots,u_k)$ denotes the linear span of vectors $u_1,\dots,u_k\in U$. I was thinking along these ...
4
votes
1answer
58 views

How to compute Nakayama functor explicitly?

I am reading the book Elements of representation theory of associative algebras, volume 1. I have some questions related to the Nakayama functor. On page 113-114 of the book, the Auslander-Reiten ...
2
votes
2answers
38 views

On the eigenvalues of a linear transformation $\tau$ such that $\tau^3 = \mathrm{id}$

I am reading the book on representation theory by Fulton and Harris in GTM. I came across this paragraph. [..] we will start our analysis of an arbitrary representation $W$ of $S_3$ by looking ...
1
vote
0answers
20 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 ...
2
votes
2answers
40 views

On the proof of Schur's lemma in Fulton & Harris

I'm reading the book on representation theory by Fulton and Harris. I'm stuck with the proof of Schur's Lemma 1.7: Schur's lemma 1.7 If $V$ and $W$ are irreducible representation of $G$ and ...
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 ...
2
votes
1answer
59 views

Question about minimal projective presentations of a module.

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1 . On page 108, line 11-14, there is a claim: If $P_0^{t}\to P_1^{t} \to TrM \to 0$ is not a minimal ...
0
votes
1answer
35 views

Are all of the irreducible representations of (any) symmetric group over $\mathbb{C}$ also irreducible over a finite splitting field.

This is probably an incredibly stupid question, but I'm a novice to representation theory and finite field theory, as I've just been introduced to these concepts, so all I really need is confirmation ...
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. ...
13
votes
1answer
196 views

Writing a group element as $ghg^{-1} h^{-1}$ and as $g^2 h^2$

I recently read the elegant paper Generalized Frobenius Schur Numbers, by Bump and Ginzburg, which I learned about here. The results in this paper imply the following: Let $G$ be a finite group ...
3
votes
2answers
51 views

$\mathbb{C}[G]$-module homomorphism on finite dimensional modules and finite groups

Nice to meet you folks! I'm currently a grad student reviewing some representation theory of finite groups for prelims next year, and I'm stuck proving a simple statement. Translating the question ...
4
votes
1answer
169 views

Geometric algebra approach to Lorentz group representations

Background: Let $\Lambda$ be the Lorentz transformation parameterized by the asymmetric real matrix $w_{\mu \nu}$. That is, let $\Lambda = \exp(\frac{w_{\mu \nu}}{2}J^{\mu \nu})$, where $(J^{\mu ...
1
vote
1answer
19 views

write representation as sum of irreducible representations

Given the representation $\rho: \mathbb{Z}/3\mathbb{Z} \rightarrow GL_2(\mathbb{C})$ by $1\rightarrow \left( \begin{array}{ccc} -1 & -1 \\ 1 & 0\\ \end{array} \right)$. I have to write this ...
3
votes
1answer
72 views

general representation theorem for bilinear forms

I am interested in representation theorems for bilinear forms, that go beyond treatment of bounded or even coercive bilinear forms. Whilst I am thankful for any references regarding the topic ...
1
vote
0answers
30 views

Representation of Homogeneous vectorbundle = Induced representation

Hello friends of mathematics :) I have a question about the induced representation. Suppose $G$ is a group and $H$ a subgroup of $G$. Suppose $\rho$ is a representation of $H$ on the vectorspace $V$, ...
1
vote
0answers
26 views

Sum of squares of the degrees of irreducible representations equals order of group (positive characteristic case) [duplicate]

Suppose $K$ is a splitting field for a finite group $G$ such that $p = \mathrm{char} K >0$ and $p \nmid |G|$. Let $\{\rho_1, \ldots, \rho_s\}$ be the set of all irreducible representations (up to ...
1
vote
2answers
37 views

Group action on vector space of all functions G to $\mathbb{C}$

I have a simple question about this following action: Let $L(G)$ be the vector space of all functions from $G$ to $\mathbb{C}$. Define an action of $G$ on $L(G)$ by $$(\sigma f)(\tau) = f(\sigma ...
4
votes
0answers
54 views

Galois representations and normal bases

I am not very familiar with the theory of Galois representations, but I do know a bit about both Galois theory and representation theory. Recently I learned about the notion of a normal basis for a ...
1
vote
0answers
47 views

What to take from representation of $S_d$?

I am reading about group representations, and books I read all contain the representation theory for symmetric groups $S_d$. However none of them presents the material in a friendly way. After reading ...
4
votes
0answers
88 views

An algebraic algorithm for finding inverses in the group algebra

This is an extension to my earlier question. Is there a purely algebraic algorithm to find inverses in the group algebra? For example, in the group algebra $\mathbb{C}S_{4}$, how would one go about ...
10
votes
2answers
115 views

Path Algebra for Categories

For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", ...
1
vote
2answers
133 views

Describing all $\rho$-invariant inner products

Let $z$ satisfying the equation $z^3=1$ be a generator of the cyclic group $\mathbb{Z}_3= \{ 1 , z,z^2 \}$. You are given that $\rho : \mathbb{Z}_3 \to GL(\mathbb{C}^2)$ defined by $$\rho(z) = ...
2
votes
1answer
46 views

Submodules of tensor representations

Let $V$ be a finite dimensional vector space over a field and $T$ the tensor algebra $T=\bigoplus_{n\geq 0} T_n,$ where $T_n=V^{\otimes n}$. It's easy to see that $T$ can be viewed as a ...
4
votes
2answers
154 views

Two non-isomorphic groups with the same complex character table

Could you give me an example of two non-isomorphic groups with the same complex character table?
7
votes
1answer
252 views

Do these two sets of matrices form groups?

Stimulated by some Physics backgrounds, consider the following two sets of matrices. Notations and definitions:Let $A,B$ be two complex $n\times n$ matrices, then $\left [ A,B \right ...
1
vote
1answer
40 views

Endomorphisms of Simple A-modules where A is a Complex algebra

Suppose $\underset{=}{\phi} \in End_A S$ is an isomorphism and $S$ is a simple (finite-dimensional?) $A$-module and $A$ is a simple $\mathbb C$-algebra. Then... must we have ...
1
vote
1answer
65 views

Exercise 2.8 M.Isaacs' Character theory of finite groups

I'm a starter at character theory. I'm trying to do this exercise: (2.8) Let $\chi$ be a faithful character of a group $G$. Show that $H\subseteq G $ is abelian if and only if every irreducible ...
0
votes
1answer
39 views

Martin Isaacs's exercise 3.7 (character theory of finite groups)

I would need some help with this exercise: Let $\chi\in{Irr(G)}$ be faithful, and suppose $\chi(1)=p^a$ for some prime p. Let $P\in{Syl_{p}(G)}$, and suppose that $C_{G}(P)\nsubseteq{P}$. Show that ...
4
votes
2answers
79 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 ...
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 ...
4
votes
1answer
107 views

Dynkin diagram automorphisms and weights

Let $\sigma$ be a nontrivial Dynkin diagram automorphism of a finite-dimensional complex simple Lie algebra $\frak g$ (of type A, D or E) and let $\frak h$ be a Cartan subalgebra of $\frak g$. Let $I$ ...
0
votes
0answers
43 views

Martin Isaacs's exercise 3.4 (character theory of finite groups)

I need some help with this: Let $G$ be a simple group and suppose $\chi\in{Irr(G)}$ with $\chi(1)=p$, a prime. Show that a Sylow $p-$subgroup of G has order p. Thanks a lot in advance.
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 ...
1
vote
1answer
41 views

Martin Isaacs's exercise 3.6 (character theory of finite groups)

I'm trying to solve this exercise, can anyone help me? Let $G$ be a p-group, and suppose $\chi\in{Irr(G)}$. Show that $\chi(1)^2$ divides $|G:Z(\chi)|$ Thanks a lot.
1
vote
1answer
49 views

Martin Isaacs's exercise 3.5 (character theory of finite groups)

I need some help with this exercise: Suppose $A\subseteq{G}$ is abelian, and $|G:A|$ is a prime power. Show that $G'\lt{G}$ Thank you very much in advance.
10
votes
5answers
186 views

Applications of Character Theory

Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, , Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character ...
0
votes
1answer
45 views

Specific question on Sn modules

Let $L_{-1}$ denote the 1-dimensional sign-representation of the symmetric group $S_n$ and V the standard $(n - 1)$-dimensional module for $S_n$. How to prove that V and $V \otimes L_{-1}$ are not ...
3
votes
3answers
53 views

Finding all submodules of G-modules

Let V; W be irreducible G-modules that are not isomorphic to each other. How to prove that the only G-submodules of M:= $V \oplus W$, other than $0$ and M itself, are $V = V \oplus  0$ and $W = 0 ...

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