Tagged Questions

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Show that the homomorphism $\lambda: k[X] \to End_k(V) : p \mapsto p(A)$ corresponding to the $k[X]$-module strucutre of $V$ has a nontrivial kernel.

$\DeclareMathOperator{\End}{End}$ I'm trying to show that: Show that the homomorphism $\lambda: k[X] \to \End_k(V) : p \mapsto p(A)$ corresponding to the $k[X]$-module strucutre of $V$ as in (see ...
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Failure of the Krull-Schmidt Theorem?

Theorem 1.19 of Representation Theory of Finite Groups: Algebra and Arithmetic is the Krull-Schmidt theorem, which I screenshotted and uploaded it here, I don't have any problem with this theorem and ...
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Why does $\rho_{\mathbf{Z}^t\otimes P}=\rho_R$ imply the isomorphsim of $\mathbf{Z}^t\otimes P\cong R$?

so I have happened upon a thesis regarding the calculations of various $\mathbf{Z}D_6$ modules and their isomorphisms and came across a technique which is bothering me. Let me give an example. Let ...
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Consider $\mathbf{Z}G$, $G$ finite. If the characters of two $\mathbf{Z}G$-modules are equal, does it follow that the modules are isomorphic?

So I have recently started to delve into integral representation theory and I was wondering if a particularly useful theorem survives the transition to integral rep theory. Basically, suppose we have ...
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Question concerning a list sorting problem

I have the following question: Let $a,b,c,d$ be four natural numbers with $a \leq b$ and $c\leq d$. I have written a program that produces a list, which has as entries all 2-tuples $(x,y)$ with ...
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Associated idempotents

Let $e$ and $f$ be elements of an associative algebra $A$. We say $e$ and $f$ are associated if there exist elements $x, y \in A$ such that: $$e = xy, f = yx.$$ My teacher said it is an easy exercise ...
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In a semi-simple module, any submodule is a direct factor?

I need help understanding the following : In a semi-simple module, any submodule is a direct factor (this is sometimes taken as the definition of semi-simple) (i) How is this equivalent to the ...
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On the indecomposable decomposition of the reduction of an integral representation

Here is a problem I have been grappling with all day. I started out thinking it might be true but am now inclined to believe it is false, and would like to see a counterexample. Suppose $G$ is a ...
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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 ...
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Find radical of a quiver representation

How to find the radical of an acyclic quiver and without relations? what is the recipe? For example suppose $Q$ is a quiver with two vertices and two arrows from vertex $2$ to vertex $1$. Now suppose ...
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Representation theory& module

$V$ is a left $R$ module, how do you understand the ring homomorphism $$\rho_{V}:R \to End_Z(V)$$ I know that it is like a group acting on sets, but it is very easy to understand like a group $S_n$ ...
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The sign representation of the Symmetric Group

I am currently trying to learn some of the basics of Representation Theory through Sagan's The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. On page 11, after ...
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The embedding $L^2(\Gamma(N)\backslash\text{SL}_2(\mathbb{R})) \hookrightarrow L^2(\text{SL}_2(\mathbb{Q})\backslash \text{SL}_2(\mathbb{A})))$?

Let $\mathbb{A}$ be the ring of adeles. Let $N$ be a natural number. Let $\Gamma(N)=\ker(\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})))$. I saw the following embedding of ...
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What are the consequences of presentation of an algebra by generators and relations?

Let $A$ be a finite dimensional associative $K$-algebra, where $K$ is a field. I wonder how the presentation of $A$ by generators and relations helps in the study of structure of the algebra ...
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Definition/existence/uniqueness of a minimal projective resolution

I'm reading Dave Benson's book "Representations and Cohomology," Volume I, and I'm trying to understand the following discussion on page $32$ in which he introduces the notion of a minimal projective ...
82 views

Characters of a faithful irreducible Module for an element in the centre

Basically here is my questions. We have a character $\chi$ which is faithful and irreducible of a group $G$. we have an element $g$ which i needs to show belongs to the centre $Z(G)$, i.e. $gh=hg$ for ...
99 views

Jordan-Holder theorem for modules?

Let $A$ be a finite dimensional algebra over some field $k$. I think from Jordan-Holder Theorem, one might be able to claim that every simple $A-$module occurs in the series (by this I mean it is ...
107 views

Modules and submodules

Let $G=S_n = Sym_n$ be the symmetric group and $V$ a vector space with basis $\{v_1,...,v_n\}$, then $V$ is a module with action defined by $g$. $v_i$=$v_{g(i)}$ for 1$\leq$i $\leq$ n and extending ...
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Prove that every 2 dimensional FG-Module with gh not equal to hg is irreducible.

Basically the questions is as follows. Suppose that $V$ is a 2-Dimensional $FG$-Module where F=The complex numbers and that there exists $g,h$ elements of $G$ and $v$ an element in $V$ such that ...
193 views

Representation theory of infinite groups?

I am familiar with the representation theory of finite groups (at least of the symmetric groups over the field of complexes) And I know that the group algebra of an infinite group is not semisimpe ...
99 views

Representation theory approach VS Module theory approach?

Given an associative algebra $A$, there is a correspondence between representations of $A$ and left $A-$ modules. Thus, one can study the representation theory of an associative algebra via its left ...
137 views

What is a regular FG-module?

What is meant by a regular $FG$-module. $G$ is a group and I believe $F$ is supposed to be a field. I'm completely confused by this concept on a question sheet and I can find lots of uses of the ...
76 views

The annihilator of finitely generated modules over PID

Let $R$ be a principal ideal domain. Let $M$ be a finitely generated $R$-module. Suppose there exists prime ideal $p$ and integer $i$ such that $p^i=\operatorname{Ann}(M)$. Then prove: (1) there ...
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Show that we have an algebra homomrohpsim

I need to show that we have an algebra homomorphism $\phi: M_n(K)\otimes_KA \simeq M_n(A)$ Where A is a K-algebra and K is some field. I suspect it's really easy but I don't know what to do. Is ...
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Proof of Clifford's theorem for modules

http://en.wikipedia.org/wiki/Clifford_theory#Proof_of_Clifford.27s_theorem I've a very easy question that I just can't seem to find the answer to. I'm self-studying so I can't ask anyone else. ...
50 views

Any submodule U of V such that the module V/U is completely reducible must contain the radical?

Want to prove: Given an $R$-Module $V$ and $rad(V)$ which I define to be the intersection of all maximal submodules of $V$. I want to show that if, for some submodule $U$ of $V$, we have $V/U$ ...
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Regular module endomorphisms into itself

Let $k$ be a field and let $A$ be an algebra over $k$. Denote by $End_A (A)$ the set of all endomorphisms of the regular $A$-module $A$ into itself. Fix $a \in A$, and define the A-module homomorphism ...
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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 ...
$\mathbb{C}[G]$-module homomorphism on finite dimensional modules and finite groups
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 ...