tags: - colorclass/functional analysis ---Schur’s Lemma is a fundamental result in the representation theory of groups, especially within the context of linear representations of finite groups and Lie groups. It plays a pivotal role in understanding the structure of representations and has significant implications for the theory of group representations and modules over algebra. The lemma is named after Issai Schur, a mathematician who made profound contributions to group theory and algebra.
Statement of Schur’s Lemma
There are two parts to Schur’s Lemma, each addressing different aspects of irreducible representations:
1. First Part: If and are vector spaces over a field , and is a linear map between irreducible representations of a group that is also a -module homomorphism (meaning it respects the group action), then is either an isomorphism or the zero map. This part of the lemma implies that there are no “non-trivial” homomorphisms between distinct irreducible representations.
2. Second Part: If is an endomorphism of an irreducible representation of a group (meaning is a linear map from to itself that commutes with the group action), then is a scalar multiple of the identity map. In other words, all endomorphisms of an irreducible representation are trivial in the sense that they can only stretch or shrink vectors but cannot change their directions relative to one another.
Implications and Applications
- Simplicity and Structure: Schur’s Lemma underscores the simplicity of irreducible representations, showing that their internal structure, in terms of endomorphisms, is minimal. This result is crucial for classifying irreducible representations, especially over algebraically closed fields like the complex numbers.
- Orthogonality of Characters: In character theory, Schur’s Lemma implies the orthogonality of characters of distinct irreducible representations, a key property used in the decomposition of regular representations and in the derivation of character formulas.
- Centralizers of Representations: The second part of Schur’s Lemma provides information about the center of the group algebra, especially concerning elements that act as scalars on irreducible representations. This has implications for understanding the structure of the algebra associated with a group.
- Representation of Abelian Groups: For abelian groups, where every element constitutes its own conjugacy class, Schur’s Lemma implies that every irreducible representation is one-dimensional (over the complex numbers). This is because the representation commutes with the action of every group element, thus must be scalar by Schur’s Lemma.
Schur’s Lemma is a cornerstone in the edifice of representation theory, offering essential insights into the nature of representations and the algebraic structures underlying groups. Its applications permeate theoretical physics, algebraic geometry, number theory, and beyond, reflecting the deep interconnections between algebra and other areas of mathematics and science.