Last update: 17 September 2013
An algebra is a vector space over with a multiplication that is associative, distributive, has an identity and satisfies the following equation An is a vector space over with an which satisfies for all and We shall use the words module and representation interchangeably.
A module is indecomposable if there do not exist non zero and such that A module is irreducible or simple if the only submodules of are the zero module and itself. A module is semisimple if it is the direct sum of simple submodules.
An algebra is simple if the only ideals of are the zero ideal and itself. The radical of an algebra is the intersection of all the maximal left ideals of An algebra is semisimple if all its modules are semisimple. An algebra is Artinian if every decreasing sequence of left ideals of stabilizes, that is for every chain of left ideals of there is an integer such that for all
The following statements follow directly from the definitions.
Let be an algebra.
|(a)||Every irreducible is indecomposable.|
|(b)||The algebra is semisimple if and only if every indecomposable is irreducible.|
The proofs of the following statements are more involved and can be found in [Bou1958] Chpt. VIII, §6, and §5,
|(a)||If is an Artinian algebra then the radical of is the largest nilpotent ideal of|
|(b)||An algebra is semisimple if and only if is Artinian and|
|(c)||Every semisimple algebra is a direct sum of simple algebras.|
The case when is not necessarily semisimple is often called modular representation theory. Let be an A composition series of is a chain such that, for each the modules are irreducible. The irreducible modules 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 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 More precisely, the submodule lattice of is the lattice defined by the submodules of with the order relations given by inclusions of submodules. The composition series are maximal chains in this lattice.
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..