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
1answer
141 views

action of orthogonal group on the space of antisymmetric bilinear forms

What is the natural action of orthogonal group on the space of antisymmetric bilinear forms.
5
votes
1answer
91 views

irreducible representation of a group

Reduced group $C^\ast$-algebra of group $G$ is defined to be $G^*_{r}(G)=\overline{\lambda(L^1(G))}$ where $\lambda$ is left regular representation. My question is how to get a irreducible ...
0
votes
1answer
218 views

Real representations of Lie algebra $\mathfrak{so}(3)$

How does one construct an $n$-dimensional, irreducible, real-valued and non-zero representation of the three generators of the Lie algebra $\mathfrak{so}(3)$ for a given value of $n$?
1
vote
3answers
291 views

Show $U \otimes V$ is an irreducible G-module

Let $G$ is some group and $U$ is an irriducible $G$-module over the complex numbers. Now if $V$ is a $G$-module of dimension 1, I would like to prove $U \otimes V$ is an irriducible $G$-module. My ...
2
votes
1answer
47 views

Meaning of representation

In mathematics, what does it mean by "one object is represented by another object"? Here is my guess. Given a mapping $f: X\to Y$. If $f$ preserves some structures when mapping from $X$ to $Y$, then ...
8
votes
3answers
2k views

Complex finite dimensional irreducible representation of abelian group

I'm supposed to show that each Complex finite dimensional irreducible representation of an abelian group is one dimensional. For any map $\phi: V \rightarrow V$ it holds that $\phi(\rho(g)v) = ...
3
votes
2answers
297 views

irreducible representation of non-abelian p-group

Can someone help with the following problem? Let $G$ be a non-abelian group of prime-power order $p^n$ and $E$ be an irreducible $G$-space over $\mathbb{C}$ giving a faithful representation of $G$. ...
2
votes
1answer
101 views

Induced representation, isomorphism between vectorspaces

I want to prove the following statement: The mapping $Ind_H^G{V}\rightarrow V^m: \psi\mapsto(\psi(g_1),\ldots,\psi(g_m))$ is a isomorphism. Here is $g_1,\ldots,g_m$ a representing system of $G/H$ ...
0
votes
1answer
49 views

Why $A/\operatorname{rad}A$ is generated by $e_a$?

Let $A$ be an algebra over an algebraically field $K$ and $(Q_A)_0$ be its ordinary quiver. Let $\{e_a \mid a \in (Q_A)_0\}$. Then $\{e_a \mid a \in (Q_A)_0\}$ is a complete set of primitive ...
0
votes
2answers
125 views

How to show that $A=(A/\operatorname{rad}A)\oplus \operatorname{rad}A$ using Wedderburn-Malcev theorem?

Let $A$ be a $K$-algebra and $K$ an algebraically closed field. How to show that $A=(A/\operatorname{rad}A)\oplus \operatorname{rad}A$ using Wedderburn-Malcev theorem? Thank you very much. ...
8
votes
1answer
167 views

Trivial summand of a representation's symmetric power

The following comes from Exercise 13.17 of Fulton and Harris's book, Representation Theory: A First Course. Let $V$ denote the standard representation of $\mathfrak{sl}_3\mathbb{C}$, with weights ...
1
vote
1answer
122 views

Irreducibility of $Sym^2$ and $\Lambda ^ 2$ representations

I'm given a representation $\Pi : \mathrm{Gl}(n,\mathbb{C}) \rightarrow \mathrm{Aut}(\mathrm{Mat}_{n\times n}(\mathbb{C}))$ by $\Pi(g) = gXg^T$ (does it have a name?). Then the representations ...
2
votes
1answer
61 views

Why $\operatorname{Hom}_A(e_jA, e_iA) \cong \operatorname{Hom}_A(e_jA, e_i\text{rad}A)$?

On line 6 of the proof of Corollary 3.4 of page 62 of the book Elements of Representation Theory of Associative Algebras, Volume 1, it is said that $\operatorname{Hom}_A(e_jA, e_iA) \cong ...
1
vote
1answer
76 views

Question about the connectivity of the ordinary quiver of a connected algebra.

I am reading the book Elements of Representation Theory of Associative Algebras, Volume 1. I have some questions about the connectivity of the ordinary quiver of a connected algebra. On page 61, ...
0
votes
1answer
158 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
306 views

A quesion in Fulton & Harris book “representation theory a first course”

In Section 11.2 A little plethysm, it discusses the tensor product of two different representations of $sl_2\mathbb{C}$. It says "If $V=\bigoplus V_{\alpha}$ and $W=\bigoplus W_{\beta}$ then ...
0
votes
2answers
58 views

Mapping from symmetric power to a lower symmetric power

This may be a dumb question, but what are the surjective maps $$f_n:\operatorname{Sym}^n(V)\to \operatorname{Sym}^{n-2}(V),$$ where Sym$^n$ denotes the $n$-th symmetric power of $V$? Wouldn't it just ...
1
vote
1answer
92 views

Embedding of $PGL_n\mathbb{C}$ and friends

I would like the find an embedding/faithful representation from the projective linear group $PGL_n\mathbb{C}\to GL_m\mathbb{C}$ for some $m$, and likewise for the other projective groups ...
5
votes
1answer
925 views

Intuition behind Maschke's theorem

I'm an undergraduate learning about group representations and Young tableaux, and have came across Maschke's theorem stating; If $G$ is a finite group and $F$ is a field who's characteristic does ...
15
votes
2answers
2k views

Is $A \times B$ the same as $A \oplus B$?

When $A, B$ are $K$-modules, then $A \times B$ is the same as $A \oplus B$. Let $A, B$ be two $K$-algebras, where $K$ is a field. Is $A \times B$ the same as $A \oplus B$? Thank you very much. ...
1
vote
0answers
186 views

Regular representation is indecomposable for characteristic $p$

Suppose we consider representations $V$ of a $p$-group $P$ over a field of characteristic $p$. If $V$ is an irreducible character, then it can be shown that $V$ is the trivial representation. ...
0
votes
1answer
120 views

Eigenvectors of algebraic group representation

In a paper of Kollar and Szabo there is a lemma (Lemma $1$) in which the following terminology is used: "Every representation $H\rightarrow GL(n,K)$ has an $H$-eigenvector" ($H$ is an algebraic ...
4
votes
0answers
167 views

How to extend a character on the maximal torus to the borel subgroup

Given a complex Lie group $G$ and a maximal torus $T$ with associated Borel subgroup $B$ and a character $\lambda : T \to U(1)$ what is the canonical extension to the Borel subgroup? The only thing ...
3
votes
1answer
116 views

Facts about induced representations

can i have help with induced representations? I find the definition of $Ind_H^GV$ difficult but i can think about it formaly. But then i want to proof the follwoing facts: ...
0
votes
1answer
35 views

How can we show that $X_A \cong T_e(Y_B)$?

I am reading the book Elements of representation theory of associative algebras 1. On Line 7 of page 37 (I attached this page below), it is said that $X_A \cong T_e(Y_B)$ since the diagram above line ...
3
votes
1answer
59 views

How to show that $Pe\otimes_B eA \cong P$?

Let $e$ be an idempotent of $A$, where $A$ is an algebra. Let $B=eAe$ and $P$ be a projective right $A$-module. How to show that $Pe\otimes_B eA \cong P$? I think that $Pe\otimes_B eAe \cong P$. But ...
1
vote
1answer
347 views

Is cyclic modules $=$ simple modules?

Let $A$ be an algebra with identity $1$ and $N$ be a right module of $A$ generated by $n_1 \in N$. That is $N=n_1A$. Is $N$ a simple module? I think that maybe this is not true. Let $N=A$ and suppose ...
0
votes
1answer
28 views

Is $eA$ simple?

Let $A$ be an algebra and $e$ be a primitive idempotent of $A$. We know that $eA$ is indecomposable as a right $A$-module. Is $eA$ a simple right $A$-module? Thank you very much.
0
votes
1answer
23 views

Simple modules over $K\times K \times \cdots \times K$.

Let $K$ be an algebraic closed field. Let $M$ be a simple module over $K\times K \times \cdots \times K$ ($n$ copies of $K$). If $n=1$, then $M \cong K$ and $\dim M =1$. If $n\geq 2$, is $\dim M =1$? ...
8
votes
1answer
97 views

Wedge pure product

Let $V$ be a vector space of $\dim n$ over $K$. Let $P$ be the set of all pure products of the form $v_1 \bigwedge v_2$. How to prove that there is a one-one correspondence between the one dimensional ...
1
vote
2answers
79 views

Induced Module by S3

Let $S_3=\{1,x,x^2,y,xy,x^{2}y | x^3=1, y^2=1, xy=yx^2\}$ be the permutation group $S_3$. Let $H=\{1,x,x^2 \}\le S_3$ . If $L$ is the trivial one-dimensional H-module, then how to show that ...
1
vote
1answer
284 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 ...
1
vote
0answers
153 views

Exterior and symmetric powers of $\mathfrak{sl}(4,\mathbb{C})$ representation

I am taking a course on representation theory, and going through Lecture 15 of Fulton and Harris's Representation Theory. One of the topics we're currently covering is the example of ...
1
vote
0answers
70 views

Projection map $\text{Sym}^2(\text{Sym}^3V)\to \text{Sym}^2V$ viewed as a Hessian

Exercises 11.21 and 11.22 in Fulton's Representation Theory are the following: Let $V$ be the standard representation of $\mathfrak{sl}_2\mathbb{C}$. The projection map from ...
3
votes
0answers
167 views

Mackey functor structure on equivariant homotopy groups

I have read that the equivariant stable homotopy groups $\pi_n^{-}(X)=\pi_n(X^{-}) $ of a $G$-space or $G$-spectrum $X$ have a Mackey functor structure. Can somebody please explain how the covariant ...
1
vote
0answers
176 views

Show group is isomorphic to finite Heisenberg group

Show that the group $\langle x,y,z$ $|$ $z = xyx^{-1}y^{-1}$, $zx=xz$, $zy = yz$, $x^n = \mathbb{I}, $ $y^n = \mathbb{I}$, $z^n = \mathbb{I} \rangle$, $(n \in \mathbb{Z_{>0}})$ is isomorphic to ...
1
vote
1answer
84 views

Tensor algebra wedge

Let $k$ be a field and $V$ a vector space of dimension $n$. Let $P$ denote the image of the universal alternating map $V \times V \to \bigwedge^2(V)$. (Thus $P$ consists of pure products of the form ...
3
votes
1answer
266 views

Exercise 2.15 M.Isaacs' Character theory of finite groups

I'm beggining to study character theory, and i'm doing some problems from Isaacs' Character theory book. I would need some help with this one: (2.15): Let $\chi\in \operatorname{Irr}(G)$ be ...
3
votes
1answer
389 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 ...
5
votes
1answer
317 views

Character theory exercises [closed]

I'm doing the exercises from chapter 2 of M.Isaacs' Character theory of finite groups, and I'm having problems with some of them. In particular, I would need help with these ones. Thank you very much ...
4
votes
2answers
135 views

How to compute a dual of a module?

Let $A=M_2(K)$ be the algebra of all $2\times 2$ matrices over $K$. Let $e_1=\left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right)$ and $e_2=\left( \begin{matrix} 0 & 0 \\ 0 & 1 ...
3
votes
1answer
46 views

How to compute $I(i)$?

Let $A=M_2(K)$ be the algebra of all $2\times 2$ matrices over $K$. Let $e_1=\left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right)$ and $e_2=\left( \begin{matrix} 0 & 0 \\ 0 & 1 ...
3
votes
2answers
134 views

Is a surjective maps from a projective module to another projective module bijection?

Let $M$ and $N$ be projective $A$-modules. If we know that $f: M \to N$ is surjective and $g: N \to M$ is surjective, can we conclude that $M$ is isomorphic to $N$? More generally, if $M$ and $N$ are ...
2
votes
1answer
81 views

Length of modules.

Let $M, N$ be two $A$-modules. If there is a surjective $A$-map from $M$ to $N$, can we conculde that $\ell(M) \geq \ell(N)$. Here $\ell(M)$ is the number of modules in a composition series of $M$. ...
2
votes
0answers
62 views

Why $\text{top}h$ is an isomorphism?

I am reading the book Elements of representation theory of associative algebras I have a question about from Line -9 to Line -6 of page 29, the proof of Theorem 5.8. How to show that ...
4
votes
2answers
238 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 ...
0
votes
1answer
168 views

Tilting modules

The theory of tilting modules seems to be a very fruitful field. I have some questions which seem natural to me, but can be trivial or stupid for people who works with this sort of theory. 1) Do all ...
1
vote
1answer
101 views

How to show that a local finite dimensional algebra is basic from definition?

A basis algebra is an algebra $A$ such that $e_i A \not\simeq e_j A$ for any $i \neq j$, where $e_1, \ldots, e_n$ is a complete set of primitive orthogonal idempotents. A local algebra is an algebra ...
2
votes
1answer
73 views

Group representation scalar product

Let $\rho: G \rightarrow GL(V)$ be a finite dimensional complex representation of the group $G$. Show that there is an inner product on $V$ such that $G$ acts by unitary matrices. My approach so far ...
1
vote
1answer
151 views

References request: Introduction to K3 surface.

Are there some good books or survey papers about K3 surface? Thank you very much.