Last update: 17 September 2013
An algebra $A$ is a vector space over $\u2102$ with a multiplication that is associative, distributive, has an identity and satisfies the following equation $$\left(c{a}_{1}\right){a}_{2}={a}_{1}\left(c{a}_{2}\right)=c\left({a}_{1}{a}_{2}\right),\phantom{\rule{2em}{0ex}}\text{for all}\hspace{0.17em}{a}_{1},{a}_{2}\in A\hspace{0.17em}\text{and}\hspace{0.17em}c\in \u2102\text{.}$$ An $A\text{-module}$ is a vector space $M$ over $\u2102$ with an $A\text{-action}$ $$\begin{array}{ccc}A\times M& \u27f6& M\\ (a,m)& \u27fc& am,\end{array}$$ which satisfies $$\begin{array}{ccc}1m& =& m,\\ {a}_{1}\left({a}_{2}m\right)& =& \left({a}_{1}{a}_{2}\right)m,\\ ({a}_{1}+{a}_{2})m& =& {a}_{1}m+{a}_{2}m,\\ a({c}_{1}{m}_{1}+{c}_{2}{m}_{2})& =& {c}_{1}\left(a{m}_{1}\right)+{c}_{2}\left(a{m}_{2}\right)\text{.}\end{array}$$ for all $a,{a}_{1},{a}_{2}\in A,$ $m,{m}_{1},{m}_{2}\in M$ and ${c}_{1},{c}_{2}\in \u2102\text{.}$ We shall use the words module and representation interchangeably.
A module $M$ is indecomposable if there do not exist non zero $A\text{-modules}$ ${M}_{1}$ and ${M}_{2}$ such that $$M\cong {M}_{1}\oplus {M}_{2}\text{.}$$ A module $M$ is irreducible or simple if the only submodules of $M$ are the zero module $0$ and $M$ itself. A module $M$ is semisimple if it is the direct sum of simple submodules.
An algebra is simple if the only ideals of $A$ are the zero ideal $0$ and $A$ itself. The radical $\text{rad}\left(A\right)$ of an algebra $A$ is the intersection of all the maximal left ideals of $A\text{.}$ An algebra $A$ is semisimple if all its modules are semisimple. An algebra $A$ is Artinian if every decreasing sequence of left ideals of $A$ stabilizes, that is for every chain $${A}_{1}\supseteq {A}_{2}\supseteq {A}_{3}\supseteq \cdots $$ of left ideals of $A$ there is an integer $m$ such that ${A}_{i}={A}_{m}$ for all $i\ge m\text{.}$
The following statements follow directly from the definitions.
Let $A$ be an algebra.
(a) | Every irreducible $A\text{-module}$ is indecomposable. |
(b) | The algebra $A$ is semisimple if and only if every indecomposable $A\text{-module}$ is irreducible. |
The proofs of the following statements are more involved and can be found in [Bou1958] Chpt. VIII, §6, ${n}^{o}4$ and §5, ${n}^{o}3\text{.}$
Theorem A1.1.
(a) | If $A$ is an Artinian algebra then the radical of $A$ is the largest nilpotent ideal of $A\text{.}$ |
(b) | An algebra $A$ is semisimple if and only if $A$ is Artinian and $\text{rad}\left(A\right)=0\text{.}$ |
(c) | Every semisimple algebra is a direct sum of simple algebras. |
The case when $A$ is not necessarily semisimple is often called modular representation theory. Let $M$ be an $A\text{-module.}$ A composition series of $M$ is a chain $$M={M}_{k}\supseteq {M}_{k-1}\supseteq \cdots \supseteq {M}_{1}\supseteq {M}_{0}=0,$$ such that, for each $1\le i\le k,$ the modules ${M}_{i}/{M}_{i-1}$ are irreducible. The irreducible modules ${M}_{i}/{M}_{i-1}$ are the factors of the composition series. The following theorem is proved in [CRe1988] (13.7).
Theorem A1.2. (Jordan-Hölder) If there exists a composition series for $M$ then any two composition series must have the same multiset of factors (up to module isomorphism).
An important combinatorial point of view is as follows: The analogue of the subgroup lattice of a group can be studied for any $A\text{-module}$ $M\text{.}$ More precisely, the submodule lattice $L\left(M\right)$ of $M$ is the lattice defined by the submodules of $M$ with the order relations given by inclusions of submodules. The composition series are maximal chains in this lattice.
References
All of the above results can be found in [Bou1958] Chapt. VIII and [CRe1988].
This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.
Key words and phrases. Algebraic combinatorics, representations.
Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..