In mathematics, Schur's lemma[1] is an elementary but extremely useful statement in representation theory of groups and algebras.
An important special case occurs when M = N, i.e. φ is a self-map; in particular, any element of the center of a group must act as a scalar operator (a scalar multiple of the identity) on M. The lemma is named after Issai Schur who used it to prove the 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 are due to Jacques Dixmier and Daniel Quillen.
Such a homomorphism is called a representation of G on V. A representation on V is a special case of a group action on V, but rather than permit any arbitrary bijections (permutations) of the underlying set of V, we restrict ourselves to invertible linear transformations.
Irreducible representations – like the prime numbers, or like the simple groups in group theory – are the building blocks of representation theory.
Just as we are interested in homomorphisms between groups, and in continuous maps between topological spaces, we are also interested in certain functions between representations of G. Let V and W be vector spaces, and let
In other words, we require that f commutes with the action of G. G-linear maps are the morphisms in the category of representations of G. Schur's Lemma is a theorem that describes what G-linear maps can exist between two irreducible representations of G. Theorem (Schur's Lemma): Let V and W be vector spaces; and let
finite-dimensional over an algebraically closed field and they have the same representation, let
(An eigenvalue exists for every linear transformation on a finite-dimensional vector space over an algebraically closed field.)
Then we return to the above argument, where we used the fact that a map was G-linear to conclude that the kernel is a subrepresentation, and is thus either zero or equal to all of
An important corollary of Schur's lemma follows from the observation that we can often build explicitly
-linear maps between representations by "averaging" over the action of individual group elements on some fixed linear operator.
In particular, given any irreducible representation, such objects will satisfy the assumptions of Schur's lemma, hence be scalar multiples of the identity.
More precisely: Corollary: Using the same notation from the previous theorem, let
For example, in the context of quantum information science, it is used to derive results about complex projective t-designs.
are submodules of simple modules, by definition they are either zero or equal to
is a simple module that is at most countably-dimensional, the only linear transformations of
When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is still of particular interest.
-algebra is said to be absolutely simple if its endomorphism ring is isomorphic to
, and implies the module is irreducible even over the algebraic closure of
is the universal enveloping algebra of a Lie algebra, a central character is also referred to as an infinitesimal character and the previous considerations show that if
is a class function, i.e. invariant under conjugation.
Since the set of class functions is spanned by the characters
A simple corollary of the second statement is that every complex irreducible representation of an abelian group is one-dimensional.
is a complex semisimple Lie algebra, an important example of the preceding construction is the one in which
is a constant that can be computed explicitly in terms of the highest weight of
[9] The action of the Casimir element plays an important role in the proof of complete reducibility for finite-dimensional representations of semisimple Lie algebras.
[10] The one-module version of Schur's lemma admits generalizations for modules
of finite length, the following properties are equivalent: Schur's lemma cannot be reversed in general, however, since there exist modules that are not simple but whose endomorphism algebra is a division ring.
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 A5 over the finite field with three elements F3 has F3 as its endomorphism ring.