Schur's lemma

For other uses, see Schur's lemma (disambiguation).

In mathematics, Schur's lemma^[1] is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations of a group G and φ is a 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 the 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.^[2]

The condition that f is a module homomorphism means that

f(rm) = rf(m)\text{ for all }m \in M\text{ 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 a field k and the vector space M = N is a simple module of R. Then Schur's lemma says that the endomorphism ring of the module M is a division algebra over the field k. If M is finite-dimensional, this division algebra is finite-dimensional. If k is the field of complex numbers, the only option is that this division algebra is the complex numbers. Thus the endomorphism ring of the module M is "as small as possible". In other words, the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity.

This holds more generally for any algebra R over an algebraically closed field k and for any simple module M that is at most countably-dimensional: the only linear transformations of M that commute with all transformations coming from R are scalar multiples of the identity.

When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is still of particular interest. A simple module over k-algebra is said to be absolutely simple if its endomorphism ring is isomorphic to k. This is in general stronger than being irreducible over the field k, and implies the module is irreducible even over the algebraic closure of k.

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,

\text{if }gV\subseteq V\text{ for all }g\text{ in }G,\text{ then either }V=0\text{ 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. If G is not irreducible, then this is not true. For example, if one takes the subgroup D of diagonal matrices inside of GL(n,C), then the center of D is D, which contains non scalar matrices. As a simple corollary, every complex irreducible representation of Abelian groups is one-dimensional.

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, and so cannot exist over semi-simple rings such as the complex group ring of a finite group. However, even over the ring of integers, the module of rational numbers has an endomorphism ring that is a division ring, specifically the field of rational numbers. Even for group rings, there are examples when the characteristic of the field divides the order of the group: the Jacobson radical of the projective cover of the one-dimensional representation of the alternating group on five points over the field with three elements has the field with three elements as its endomorphism ring.

Notes

↑ Issai Schur (1905) "Neue Begründung der Theorie der Gruppencharaktere" (New foundation for the theory of group characters), Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, pages 406-432.
↑ Lam (2001), p. 33.

References