Schur's Lemmas and Intertwining Maps

1. The Intertwining Operator: A Map That Respects Symmetry

Before stating the lemmas, we must define the object they act upon: the intertwining operator, also known as a G-homomorphism or G-linear map.

Let $(V_1,{\rho }_1)$ and $(V_2,{\rho }_2)$ be two representations of a group $G$. A linear map $u:V_1\to V_2$ is called an intertwining operator if it satisfies the following condition for all group elements $g\in G$:

This equation expresses a fundamental compatibility between the map $u$ and the group action. It means that the diagram below commutes:

The set of all such intertwining operators between $V_1$ and $V_2$ is denoted ${\text{Hom}}_G(V_1,V_2)$. Schur's lemmas tell us about the structure of this set when the representations are irreducible.

2. Schur's Lemmas: The Core Theorems

Schur's lemmas come in two parts, dealing with maps between inequivalent irreps and maps within a single irrep.

Let's summarize and formalize the results from the reading. Let $(V_1,{\rho }_1)$ and $(V_2,{\rho }_2)$ be irreducible representations of a group $G$, and let $u:V_1\to V_2$ be an intertwining operator.

• Schur's First Lemma (for maps between different irreps):
  The map $u$ is either the zero map ($u=0$) or an isomorphism.

  • Proof Sketch: As the reading shows, $\ker (u)$ is an invariant subspace of $V_1$, and $\text{Im}(u)$ is an invariant subspace of $V_2$. Since ${\rho }_1$ and ${\rho }_2$ are irreducible:
    • $\ker (u)$ can only be $\{0\}$ or $V_1$. If $\ker (u)=V_1$, then $u=0$.
    • $\text{Im}(u)$ can only be $\{0\}$ or $V_2$. If $\text{Im}(u)=\{0\}$, then $u=0$.
    • The only other possibility is $\ker (u)=\{0\}$ (so $u$ is injective) and $\text{Im}(u)=V_2$ (so $u$ is surjective). Together, this means $u$ is an isomorphism.

• Corollary/Schur's Second Lemma: If ${\rho }_1$ and ${\rho }_2$ are inequivalent irreducible representations, then the only intertwining operator between them is the zero map. That is, ${\text{Hom}}_G(V_1,V_2)=\{0\}$. This follows directly, since if they are inequivalent, no isomorphism between them can exist by definition.

• Schur's Second Lemma (for maps within a single irrep):
  If we have an irreducible representation $(V,\rho )$ on a complex vector space $V$, and $u:V\to V$ is an intertwining operator (i.e., it commutes with all $\rho (g)$), then $u$ must be a scalar multiple of the identity operator.

  $$u=\lambda I_V,\text{for some }\lambda \in \mathbb{C}$$
  • Proof Sketch: Since we are working over the complex numbers, the linear operator $u$ is guaranteed to have at least one eigenvalue, $\lambda$. Consider the operator $u^{\prime }=u-\lambda I_V$. This operator also commutes with all $\rho (g)$. The kernel of $u^{\prime }$ is, by definition, the eigenspace of $u$ corresponding to $\lambda$, so $\ker (u^{\prime })$ is not the zero subspace. Since $\ker (u^{\prime })$ is a non-trivial invariant subspace of an irreducible representation, it must be the entire space $V$. This implies $u^{\prime }=0$, and therefore $u=\lambda I_V$.

These lemmas are incredibly restrictive. They state that irreps are so "atomic" that they are either completely unrelated (no non-zero maps between them) or, if a map exists from an irrep to itself that preserves the symmetry, it must be something as simple as scalar multiplication.

3. Physical Applications: Degeneracy and Casimir Operators

The abstract power of Schur's lemmas becomes concrete when applied to quantum mechanics. Given your background in QFT and condensed matter, these applications will be particularly relevant.

Application 1: Operators Commuting with a Symmetry

Recall from our last lesson that the degenerate eigenfunctions of a Hamiltonian $H$ at energy $E$ form a vector space $V_E$ which provides a basis for an irrep, ${\rho }_E$, of the Hamiltonian's symmetry group $G$.

Now, consider another physical observable, represented by a Hermitian operator $O$. If this observable is also invariant under the symmetry group $G$, then $O$ must commute with all the symmetry operators: $[O,{\rho }_E(g)]=0$ for all $g\in G$.

This commutation relation is precisely the condition for $O$ to be an intertwining operator on the space $V_E$. By Schur's second lemma, the action of $O$ when restricted to the subspace of degenerate states $V_E$ must be proportional to the identity.

This has a profound physical consequence: all states within a single irreducible energy multiplet are eigenstates of $O$ with the same eigenvalue ${\lambda }_E$. The symmetry forbids $O$ from distinguishing between the degenerate states. This provides a deep reason for why particles in a multiplet share certain quantum numbers.

Application 2: The Casimir Operator

A more advanced and elegant application involves Casimir operators, which are central to the study of Lie algebras.

A Casimir operator $\Omega$ is an operator constructed from the generators of a Lie algebra that commutes with all of them. The archetypal example is the total angular momentum operator ${\mathbf{J}}^2=J_x^2+J_y^2+J_z^2$, which commutes with $J_x$, $J_y$, and $J_z$, the generators of the SO(3) Lie algebra.

Since $\Omega$ commutes with all the generators, its representation $\rho (\Omega )$ will commute with the representation of every group element $\rho (g)$ (this can be shown via the exponential map). Therefore, $\rho (\Omega )$ is an intertwining operator on the representation space $V$.

If the representation $(V,\rho )$ is irreducible, Schur's lemma immediately tells us that $\rho (\Omega )$ must be a constant multiple of the identity:

This means all states in an irreducible representation are eigenstates of the Casimir operator with the same eigenvalue $C_{\rho }$. This is precisely why we can label the irreps of SO(3) by the quantum number $j$ (or spin). All $2j+1$ states in the multiplet have the same value of total angular momentum, ${\hslash }^{2}j(j+1)$, which is the eigenvalue of the Casimir operator ${\mathbf{J}}^2$. The Casimir eigenvalue serves as a unique label for the irrep itself.

The lecturer then proceeds to calculate the value of this constant and apply it to specific examples. This shows how the abstract lemma leads to concrete, computable numbers that characterize physical systems.

Conclusion

In this lesson, we've explored the profound implications of Schur's lemmas. These theorems impose a rigid structure on the relationships between irreducible representations, effectively isolating them from one another.

Here are the key takeaways:

• An intertwining operator $u$ is a linear map between representation spaces that commutes with the group action: $u\circ {\rho }_1(g)={\rho }_2(g)\circ u$.
• Schur's lemmas state that for irreducible representations:
  1. Any intertwining map between inequivalent irreps must be the zero map.
  2. Any intertwining map from a complex irrep to itself must be a scalar multiple of the identity, $u=\lambda I$.
• Physical Significance: These lemmas explain why all states in a degenerate energy multiplet have the same quantum numbers for any other operator that respects the system's symmetry. They also explain why Casimir operators, like total angular momentum ${\mathbf{J}}^2$, have a single, constant value across an entire irrep, allowing this value to serve as a label for the irrep itself.

Schur's lemmas form the theoretical bedrock for the next major tool in our arsenal: the great orthogonality theorem. While the lemmas tell us why irreps are so constrained, the orthogonality theorem will give us the practical, computational machinery to exploit these constraints. In the next lesson, we will begin to develop this machinery by introducing the concept of characters.