Schur's Lemma

Schur's Lemma is one of the most fundamental results in representation theory, providing crucial information about morphisms between irreducible representations and the structure of endomorphism algebras.

TheoremSchur's Lemma (Basic Form)

Let $V$ and $W$ be irreducible representations of a group $G$ over an algebraically closed field $\mathbb{F}$ .

Distinctness: If $V$ and $W$ are non-isomorphic, then $\text{Hom}_G(V, W) = \{0\}$
Self-maps: If $V = W$ , then every intertwining map $\phi: V \to V$ is a scalar multiple of the identity: $\text{End}_G(V) = \mathbb{F} \cdot \text{id}_V$

In other words, $\text{Hom}_G(V, W) \cong \begin{cases} \mathbb{F} & \text{if } V \cong W \\ 0 & \text{otherwise} \end{cases}$

Proof sketch: For (1), if $\phi: V \to W$ is a non-zero morphism, then $\ker(\phi)$ and $\text{im}(\phi)$ are subrepresentations. By irreducibility of $V$ and $W$ , we have $\ker(\phi) = \{0\}$ and $\text{im}(\phi) = W$ , so $\phi$ is an isomorphism.

For (2), let $\phi: V \to V$ be an intertwining map. Since $\mathbb{F}$ is algebraically closed, $\phi$ has an eigenvalue $\lambda \in \mathbb{F}$ . Then $\phi - \lambda \text{id}_V$ is an intertwining map with non-trivial kernel, hence by irreducibility must be zero. Thus $\phi = \lambda \text{id}_V$ .

ExampleApplications

For finite groups over $\mathbb{C}$ : If $\chi$ is the character of an irreducible representation, then: $\langle \chi, \chi \rangle = \frac{1}{|G|} \sum_{g \in G} |\chi(g)|^2 = 1$

Matrix representations: If $\rho: G \to GL_n(\mathbb{C})$ is irreducible and $M$ is a matrix commuting with all $\rho(g)$ , then $M = \lambda I_n$ for some $\lambda \in \mathbb{C}$ .

Multiplicity formula: In a decomposition $V = \bigoplus_i V_i^{\oplus m_i}$ where $V_i$ are distinct irreducibles, the multiplicity $m_i = \dim \text{Hom}_G(V_i, V)$ .

Remark

The algebraically closed field assumption in part (2) is essential. Over $\mathbb{R}$ , the rotation group $SO(2)$ has a two-dimensional irreducible representation (rotations of $\mathbb{R}^2$ ), and rotation by any angle provides a non-scalar endomorphism. However, over $\mathbb{C}$ , this representation becomes reducible.

TheoremSchur's Lemma (Refined Form)

For finite-dimensional representations over $\mathbb{C}$ :

If $V$ is irreducible, then $\text{End}_G(V) \cong \mathbb{C}$ is a division algebra. If additionally the field is algebraically closed and $\dim V < \infty$ , then $\text{End}_G(V) = \mathbb{C} \cdot \text{id}_V$ by the spectral theorem.

For infinite-dimensional representations, $\text{End}_G(V)$ can be a larger division algebra (e.g., quaternions for certain real representations).

Schur's Lemma underlies the orthogonality relations for characters, the decomposition of tensor products, and Frobenius reciprocity. It is the foundation upon which much of the representation theory of finite groups and compact Lie groups is built.