4
votes
1answer
62 views

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 ...
0
votes
1answer
33 views

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 ...
5
votes
1answer
89 views

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 ...
2
votes
1answer
78 views

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 ...
2
votes
1answer
34 views

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 ...
0
votes
0answers
39 views

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 $S_n$ ...
0
votes
1answer
38 views

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 ...
2
votes
0answers
27 views

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 ...
1
vote
0answers
87 views

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 ...
2
votes
1answer
109 views

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 ...
2
votes
2answers
79 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 ...
5
votes
1answer
87 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 ...
3
votes
1answer
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 ...
0
votes
0answers
91 views

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 ...
2
votes
2answers
173 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 ...
8
votes
1answer
95 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 ...
1
vote
1answer
110 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 ...
0
votes
0answers
69 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 ...
0
votes
0answers
50 views

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 ...
3
votes
1answer
74 views

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. ...
1
vote
1answer
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 $V/U$ ...
1
vote
1answer
28 views

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 ...
1
vote
0answers
79 views

Irreducible modules - semisimple algebras and endomorphism rings

Let $A$ be a finite dimensional, semi-simple $k$-algebra and $V$ and irreducible $A$-module. I am trying to prove the following claim: If $B = \text{End}_A(V^{\oplus r})$ then $$W = ...
0
votes
1answer
35 views

Show that $End_A(A)$ = {$r_a$ | $a ∈ A$}

Let $k$ be a field and let $A$ be a $k$-algebra. Denote by $End_A(A)$ the set of all $A$-homomorphisms of the regular $A$-module $A$ into itself. Fix $a ∈ A$, and define the $A$-module homomorphism $r_a ...
1
vote
0answers
230 views

Spinor representation and Clifford modules

Let $V$ be an even-dimensional real inner product space. We denote the Clifford algebra of $V$ by $C(V)$ and the spinor representation by $S$. For a finite-dimensional $\mathbb Z_2$-graded complex ...
0
votes
0answers
43 views

Multiplicity free module and centralizer of subalgebra

This is related to a previous question I posted here. Let $A$ be a semisimple algebra and $B\subset A$ be a subalgebra. Define $$C_A(B) = \{a\in A | ab=ba \quad \forall b\in B\}$$ Let $V$ and ...
3
votes
0answers
73 views

$\text{Hom}$ of irreducible modules and restrictions

This question is in reference to this paper. More specifically it is in reference to the proof of proposition 1.4 on page 8. First a defintion: Let $A$ be a semisimple finite dimensional ...
2
votes
1answer
149 views

Finding Idempotents?

I was wondering if there is a method for finding primitive idempotents of a finite dimensional algebra (over a field)? or in other words is there any way to build the complete set of primitive ...
1
vote
0answers
78 views

Representations of the algebraic group $\mathrm{GL}_n$

Let $R$ be a commutative ring and $M$ some $R$-module. How can we describe concisely an action of the algebraic group $\mathrm{GL}_{n,R}$ on $\tilde{M}$? This corresponds to a coaction from the Hopf ...
1
vote
0answers
69 views

If $S$ is a simple $\mathbb{C}G$-module and $U$ is a one-dimensional $\mathbb{C}G$-module, then $S \otimes U$ is simple

Show that if $S$ is a simple $\mathbb{C}G$-module and $U$ is a one-dimensional $\mathbb{C}G$-module, then $S \otimes U$ is simple. I've tried to prove this fact, but I think ther's a mistake ...
1
vote
1answer
58 views

Reference request for Modular representation theory

I am trying to learn modular representation theory. I would be thankful if any one tell me a good reference to start with?
2
votes
2answers
152 views

Example of a simple module which does not occur in the regular module?

Let $K$ be a field and $A$ be a $K$-algebra. I know, if $A$ is artinain algebra, then by Krull-Schmidt Theorem $A$ , as a left regular module, can be written as a direct sum of indecomposable ...
2
votes
1answer
34 views

What are modules in $\operatorname{add} T$ explicitly?

Let $A$ be a $K$-algebra and $T$ an $A$-module. The category $\operatorname{add} T$ is defined as the smallest additive subcategory of the category $\operatorname{mod} A$ (the category of all finite ...
1
vote
1answer
71 views

How to compute the ordinary quiver of $B = \operatorname{End}_A(T_{A})$?

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory by Ibrahim Assem, Daniel Simson, Andrzej Skowronski. Let $A$ be a ...
1
vote
1answer
39 views

How to show that ${}_{B}T_{A} \otimes DM \in \operatorname{Gen}({}_{B} T)$?

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory by Ibrahim Assem, Daniel Simson, Andrzej Skowronski. Let $A$ be a ...
0
votes
1answer
52 views

How to show that $DA\cong D\operatorname{Hom}_{B}(T, T) \cong DT \otimes_{B} T$?

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory by Ibrahim Assem, Daniel Simson, Andrzej Skowronski. Let $A$ be a ...
1
vote
0answers
74 views

Extending isomorphisms in the semi-simple case.

Is there some proposition saying how to extend an isomorphism of $k$-vector spaces where $k$ is a field of characteristic $p$ to an isomorphismus of $k[H]$-modules where $H$ is a group of order prime ...
1
vote
1answer
67 views

Eigenspace decomposition for semisimple module

Let's start with a prime $p$ and a group $\Delta$ of order prime to $p$. Let $M$ be a finite $\mathbb{F}_p[\Delta]$-module of order a power of $p$. I want to find a decomposition into eigenspaces ...
5
votes
1answer
87 views

If $M \simeq N$ in ${\tt stmod}(G)$ will $M \oplus \text{(proj)} \simeq N \oplus \text{(proj)}$ in ${\tt mod}(G)$?

Let $G$ be a finite group and ${\tt stmod}(G)$ the stable module category for $G$, i.e., the category whose objects are $G$-modules and whose morphisms are $G$-module homomorphisms modulo those that ...
3
votes
1answer
95 views

First group cohomology and composition factors

Let $G$ be a finite group. Let $k$ be a field ($\text{char}(k)=p>0$). Let $P(k)$ be the projective cover of $k$. Assume that for any nontrivial simple $kG$-module $M$ we have $H^1(G,M)=0$. Does it ...
2
votes
2answers
39 views

Partial cycles in projective resolutions of square-free algebra

Short version: Over a square-free algebra must every projective resolution of a simple module eventually terminate or contain a shift of itself as a direct summand? I suspect not, but have not ...
7
votes
1answer
122 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 ...
4
votes
1answer
165 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 ...
3
votes
2answers
120 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 ...
2
votes
1answer
128 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
68 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
51 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 ...
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 ...
5
votes
0answers
83 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} ...