1
vote
0answers
67 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 ...
1
vote
2answers
50 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
63 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
52 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
95 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
80 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
132 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
74 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
62 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
58 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
48 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
64 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
48 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
68 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
169 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
34 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
66 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
131 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
75 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
63 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
55 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
139 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
63 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
38 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
51 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
73 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
60 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
80 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
88 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
37 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
116 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
129 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
109 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
107 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 ...
3
votes
1answer
59 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
49 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
20 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
33 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
81 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} ...
1
vote
1answer
69 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 ...
1
vote
1answer
28 views

Questions about maximal submodules.

Let $A$ be a $K$-algebra and $M$ a right $A$-module, where $K$ is a field. Suppose that $M=C\oplus D$, where $C, D$ are right $A$-modules. If $C', D'$ are maximal right $A$-submodules of $C, D$ ...
2
votes
1answer
52 views

Characters of elements under every representation equal implies conjugacy

If $G$ is a group, suppose that for every $G$-module $V$ we have $$\chi_V(g_1)=\chi_V(g_2).$$ How can I be sure $g_1$ and $g_2$ are conjugate in $G$? Its easy to the reverse implication; ...
3
votes
2answers
104 views

Question about the radical of the Jacobson radical.

I am confused about the notation $\operatorname{rad}^2 A$. It can be considered as $\operatorname{rad}(\operatorname{rad}(A))$ or as $(\operatorname{rad}(A))^2$. Are ...
2
votes
1answer
43 views

Surjective map results in subrepresentation

I need to prove that a surjective homomorphism of finite $\mathbb{F}_p[\Delta]$-modules $$A \twoheadrightarrow B$$ results in $B$ being a subrepresenation of $A$ of the group $\Delta$ of order prime ...
1
vote
1answer
104 views

Artinian ring with zero finitistic dimension

Let $R$ be a left artinian ring with identity. Suppose $R$ contains copies of all its simple right $R$-modules. Is it true that every left $R$-module of finite projective dimension is projective (so ...
1
vote
3answers
163 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 ...
0
votes
1answer
37 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 ...