Schur's lemma

Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that that if M and N are two finite-dimensional irreducible representations of a group G and φ is linear map from M to N that commutes with the action of the group, then either φ is invertible, or φ = 0. An important special case occurs when M = N and φ is a self-map. The lemma is named after Issai Schur who used it to prove Schur orthogonality relations and develop the basics of representation theory of finite groups. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which is due to Jacques Dixmier.

Formulation in the language of modules

If M and N are two simple modules over a ring R, then any homomorphism f: M → N of R-modules is either invertible or zero. In particular, the endomorphism ring of a simple module is a division ring.

The condition that f is a module homomorphism means that

$f(rm)=rf(m)$ for all $m$ in $M$ and $r$ in $R.$

The group version is a special case of the module version, since any representation of a group G can equivalently be viewed as a module over the group ring of G.

Schur's lemma is frequently applied in the following particular case. Suppose that R is an algebra over the field C of complex numbers and M = N is a finite-dimensional module over R. Then Schur's lemma says that any endomorphism of the module M is either given by a multiplication by a non-zero scalar, or is identically zero. This can be expressed by saying that the endomorphism ring of the module M is C, that is, "as small as possible". More generally, this results holds for algebras over any algebraically closed field and for simple modules that are at most countably-dimensional.

Matrix form

Let G be a complex matrix group. This means that G is a set of square matrices of a given order n with complex entries and G is closed under matrix multiplication and inversion. Further, suppose that G is irreducible: there is no subspace V other than 0 and the whole space which is invariant under the action of G. In other words,

if $gV\subseteq V$ for all $g$ in $G$, then either $V=0$ or $V=\mathbb {C} ^{n}.$

Schur's lemma, in the special case of a single representation, says the following. If A is a complex matrix of order n that commutes with all matrices from G then A is a scalar matrix.

Generalization to non-simple modules

The one module version of Schur's lemma admits generalizations involving modules M that are not necessarily simple. They express relations between the module-theoretic properties of M and the properties of the endomorphism ring of M.

A module is said to be strongly indecomposable if its endomorphism ring is a local ring. For the important class of modules of finite length, the following properties are equivalent (Lam 2001, §19):

• A module M is indecomposable;
• M is strongly indecomposable;
• Every endomorphism of M is either nilpotent or invertible.

In general, Schur's lemma cannot be reversed: there exist modules that are not simple, yet their endomorphism algebra is a division ring. Such modules are necessarily indecomposable.

References

• David S. Dummit, Richard M. Foote. Abstract Algebra. 2nd ed., pg. 337.
• Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0