Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 17 September 2013

Appendix A

A1. Basic Representation Theory

An algebra A is a vector space over with a multiplication that is associative, distributive, has an identity and satisfies the following equation (ca1)a2= a1(ca2)= c(a1a2), for alla1, a2Aand c. An A-module is a vector space M over with an A-action A×M M (a,m) am, which satisfies 1m = m, a1(a2m) = (a1a2)m, (a1+a2)m = a1m+a2m, a(c1m1+c2m2) = c1(am1)+ c2(am2). for all a,a1,a2A, m,m1,m2M and c1,c2. We shall use the words module and representation interchangeably.

A module M is indecomposable if there do not exist non zero A-modules M1 and M2 such that MM1M2. 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 rad(A) of an algebra A is the intersection of all the maximal left ideals of A. 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 A1A2 A3 of left ideals of A there is an integer m such that Ai=Am for all im.

The following statements follow directly from the definitions.

Let A be an algebra.

(a) Every irreducible A-module is indecomposable.
(b) The algebra A is semisimple if and only if every indecomposable A-module is irreducible.

The proofs of the following statements are more involved and can be found in [Bou1958] Chpt. VIII, §6, no4 and §5, no3.

Theorem A1.1.

(a) If A is an Artinian algebra then the radical of A is the largest nilpotent ideal of A.
(b) An algebra A is semisimple if and only if A is Artinian and rad(A)=0.
(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-module. A composition series of M is a chain M=MkMk-1 M1M0=0, such that, for each 1ik, the modules Mi/Mi-1 are irreducible. The irreducible modules Mi/Mi-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-module M. More precisely, the submodule lattice L(M) 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.


All of the above results can be found in [Bou1958] Chapt. VIII and [CRe1988].

Notes and references

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..

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