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

learn more… | top users | synonyms (1)

3
votes
0answers
80 views

Mapping $G$ into its group algebra as left multiplication. Continuous?

I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove: Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
1
vote
1answer
120 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
191 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. ...
3
votes
2answers
132 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 ...
1
vote
1answer
46 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 ...
1
vote
0answers
199 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
2answers
156 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 ...
1
vote
0answers
31 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 ...
12
votes
1answer
203 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 ...
10
votes
1answer
213 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
57 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 ...
3
votes
1answer
74 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 ...
10
votes
2answers
327 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
1answer
70 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 ...
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 ...
0
votes
1answer
108 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 ...
6
votes
0answers
119 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
186 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.
3
votes
1answer
136 views

Finite groups such that $H^1(G,M)=0$ for any simple $G$-module $M$

I'm trying to understand for which finite groups $G$ the augmentation ideal of $\mathbb{F}_2G$ is generated by a single element over $\mathbb{F}_2G$. I'm reading a paper with a result that implies ...
1
vote
1answer
135 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.
0
votes
1answer
70 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
193 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 ...
3
votes
1answer
46 views

multiplicity of irreducible components of S3 modules

Let V denote the 2 dimensional irreducible standard module for $S_3$. I want to find multiplicity of each of irreducible components of $V^{\otimes ^{10}}$ , by writing the character for $V^{\otimes ...
11
votes
5answers
307 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 ...
4
votes
0answers
40 views

Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$

If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
2
votes
1answer
160 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 ...
4
votes
1answer
78 views

Does the projectively stable category have projective modules?

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1 . On page 109, the projectively stable category is defined by $$ \underline{mod} A = mod A/\mathcal{P}. $$ ...
1
vote
1answer
56 views

Image of the projection map onto an irreducible module

Let $A$ be a $K$-algebra $V$ be an $A$-module.Let $V=\bigoplus W_j$ with $W_j$ irreducible for all j .Let $M$ be an irreducible $A$-module. And $N=\Sigma \{W_i: W_i$ is isomorphic to $M\}$. Now let ...
2
votes
2answers
50 views

Group ring is not isomorphic to 2 by 2 matrices

Let k be an algebraically closed field of characteristic 0. How to prove that $M_2$(k), the k-algebra of 2 by  2 matrices over k, is not isomorphic to the group ring of any finite group G over k.
4
votes
3answers
367 views

Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$, where $G$ is abelian and $\mathbb{T}$ is the circle group.

Suppose $G$ is an abelian group and $a\in G$ and $$f:\left<a \right>\to\Bbb T$$ is a homomorphism. Can $f$ be extended to a homomorphism on $G$: $$g:G\to \Bbb T$$ ? $\Bbb T$ is the circle ...
7
votes
2answers
424 views

Algebraic geometry in representation theory?

I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, ...
2
votes
0answers
75 views

Character of half-spin representation

Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum $$\sum x_1^{\pm ...
2
votes
1answer
166 views

Computing Invariant Subspaces of Matrix Groups

Does anyone have a program written in Mathematica (or SAGE or GAP) that computes the invariant subspace lattice of a matrix group?
7
votes
1answer
355 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 ...
2
votes
1answer
130 views

properties of Sym^2 vector subspace/properties of tensor products

I have a problem: Let $V$ be an $n$-dimensional complex vector space and let $B=\{e_1,e_2,...,e_n\}$ denote the elements of a chosen basis. Let $\rho:G \to GL(V)$ be an irreducible representation. Let ...
4
votes
1answer
31 views

Harish-Chandra modules of $PSL_2(\mathbb{R})$

Let $G=PSL_2(\mathbb{R})$ and $K$ a maximal torus. Is the category of Harish-Chandra modules of $(G,K)$ equivalent to the Category of Harish-Chandra Modules of $SL_2(\mathbb{R})$ with even $K$-types? ...
0
votes
1answer
21 views

Submodules and $p$-adic numbers

I am a little bit confused about the terminology of simple $\mathbb{Q}_p[G]$ module. E.j.: If one take an $\mathbb{Z}_p[G]$ module $M$, then $pM$ is a submodule, so one can just look for ...
0
votes
1answer
34 views

$\operatorname{Res}(V+W)=\operatorname{Res}(V)+\operatorname{Res}(W)$?

if there are two $R[G]$ Modules $V,W$ and $R$ some ring, $S$ subgroup of $G$. Is the formula $$\operatorname{Res}_S (V \oplus W) = \operatorname{Res}_S (V) \oplus \operatorname{Res}_S (W) $$ true? I ...
1
vote
1answer
165 views

Representations - Tensor Product prove properties of tensor product

I have a problem: Let $V$ be an $n$-dimensional complex vector space and let $B=\{e_1,e_2,...,e_n\}$ denote the elements of a chosen basis. Let $\rho:G \to GL(V)$ be an irreducible representation. Let ...
3
votes
1answer
113 views

Braid Group of a Weyl Group

I am reading the paper Cherednik Algebras, Macdonald Polynomials, and Combinatorics by Mark Haiman. The definition (2.7) of the braid group $\mathcal{B}(W)$ seems to be the same as the definition of ...
5
votes
3answers
158 views

Are Clifford groups very *non-commutative*?

Clifford groups seem to be very non-commutative by the relation \begin{equation} \gamma_{i}\gamma_{j}=-\gamma_{j}\gamma_{i}. \end{equation} But is it really so? Can we put this degree of ...
5
votes
0answers
91 views

Deciding whether or not a class of modules is “big enough”

For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
6
votes
0answers
80 views

Duality of $Z(G)$ and $[G,G]$ in representation?

This question and its many wonderful answers illustrate many faces of the duality of $Z(G)$ and $[G,G]$, the centre/ commutator duality of a group. I was thinking about its manifestation in group ...
3
votes
2answers
141 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 ...
1
vote
1answer
80 views

Show that for a finite field $F$ and finite group $G$, $F$ is a splitting field for $FG$

Let $F=\mathbb{F}_q$ be the field of $q$ elements and $G$ be a finite group. I'm trying to show that for an irreducible $FG$-module $V$, we have $\mathrm{End}_{FG}(V)=F \cdot 1$, i.e. that $F$ is a ...
2
votes
1answer
26 views

Question about the top of a bound representation of a bound quiver.

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1. I have a on page 77. In (d) of Lemma 2.2 on Page 77, it is said that $$ L_a=\sum_{\alpha: a\to b} ...
3
votes
1answer
45 views

How to compute the weights of $\Gamma_{3,1}$ the irrep of $\mathfrak{sl}_3\Bbb C$

I am wondering about a combinatorial formula for computing the weights of the irreducible representations $\Gamma_{a,b}$ of $\mathfrak{sl}_3\Bbb C$. By $\Gamma_{a,b}$ I mean the irrep that has highest ...
3
votes
1answer
182 views

Weights versus roots

I am not sure of the difference between weights and roots. Am I correct in thinking that the weights are the eigenvalues of the action of the maximal torus on a given representation, and the roots are ...
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 $\#G$ ...
1
vote
2answers
109 views

Computing eigenvalues for $\mathrm{Sym}^2(\mathrm{Sym}^3 V))$ for $V = \Bbb C^2$

Given $V = \Bbb C^2$ the standard representation of $\mathfrak{sl}_2\Bbb C$, on page 157 of Fulton and Harris's Representation Theory they state Since $U = \mathrm{Sym}^3 V$ has eigenvalues $-3, ...