Schur's Lemma

## Schur's lemma

Let $A$ be an algebra and let $M$ be an $A$-module. Define $End A M = T ∈ End M | T a = a T for all a ∈ A .$

(Schur's lemma). Let $A$ be a be a finite dimensional algebra over an algebraically closed field $\stackrel{‾}{𝔽}$.

1. Let ${A}^{\lambda }$ be a simple $A$-module. Then ${\mathrm{End}}_{A}\left({A}^{\lambda }\right)=\stackrel{‾}{𝔽}·{\mathrm{Id}}_{{A}^{\lambda }}\text{.}$
2. If ${A}^{\lambda }$ and ${A}^{\mu }$ are nonisomorphic simple $A$-modules then ${\mathrm{Hom}}_{A}\left({A}^{\lambda },{A}^{\mu }\right)=0$.

 Proof. (b): Let $T:{A}^{\lambda }\to {A}^{\mu }$ be a nonzero $A$-module homomorphism. Since ${A}^{\lambda }$ is simple, $\mathrm{ker}T=0$ so $T$ is injective. Since ${A}^{\mu }$ is simple, $\mathrm{im}T={A}^{\mu }$ and so $T$ is surjective. So $T$ is an isomorphism. Thus we may assume that $T:{A}^{\lambda }\to {A}^{\lambda }$. (a): Since $\stackrel{‾}{𝔽}$ is algebraically closed $T$ has an eigenvector and a corresponding eigenvalue $\alpha \in \stackrel{‾}{𝔽}$. Then $T-\alpha ·\mathrm{Id}$ is not invertible. So $T-\alpha ·\mathrm{Id}=0$. So $T=\alpha ·\mathrm{Id}$. So ${\mathrm{End}}_{A}\left({A}^{\lambda }\right)=\stackrel{‾}{𝔽}·\mathrm{Id}$. $\square$

1. $M\cong {\oplus }_{\lambda \in \stackrel{ˆ}{M}}{M}_{{m}_{\lambda }}\left(\stackrel{‾}{𝔽}\right)$.
2. As an $\left(A,Z\right)$-module $M ≅ ⨁ λ ∈ M ˆ A λ ⊗ Z λ ,$ where the ${Z}^{\lambda }$, $\lambda \in \stackrel{ˆ}{M}$, are the simple $Z$-modules.

 Proof. (a): Index the components in the decomposition of $M$ by dummy variables ${\epsilon }_{i}^{\lambda }$ so that we may write $M ≅ ⨁ λ ∈ M ˆ ⨁ i = 1 m λ A λ ⊗ ε i λ .$ For each $\lambda \in \stackrel{ˆ}{M}$, $1\le i,j\le {m}_{\lambda }$, let ${\phi }_{ij}^{\lambda }:{A}^{\lambda }\otimes {\epsilon }_{j}\to {A}^{\lambda }\otimes {\epsilon }_{i}$ be the $A$-module isomorphism given by $φ i j λ m ⊗ ε j λ = m ⊗ ε i λ , for m ∈ A λ .$ By Schur's lemma, $End A M = Hom A M M ≅ Hom A ⨁ λ ⨁ j A λ ⊗ ε j λ ⨁ μ ⨁ i A μ ⊗ ε i μ ≅ ⨁ λ μ ⨁ i j δ λ μ Hom A A λ ⊗ ε j λ A μ ⊗ ε i μ ≅ ⨁ λ ⨁ i j = 1 m λ 𝔽 ‾ φ i j λ .$ Thus each element $z\in {\mathrm{End}}_{A}\left(M\right)$ can be written as $z = ∑ λ ∈ M ˆ ∑ i j = 1 m λ z i j λ φ i j , for some z i j λ ∈ 𝔽 ‾ ,$ and can be identified with an element of ${⨁}_{\lambda }{M}_{m\lambda }\left(\stackrel{‾}{𝔽}\right)$. Since ${\phi }_{ij}^{\lambda }{\phi }_{ij}^{\mu }={\delta }_{\lambda \mu }{\delta }_{jk}{\phi }_{ij}^{\lambda }$ it follows that $End A M ≅ ⨁ λ ∈ M ˆ M m λ 𝔽 ‾ .$ (b): As a vector space, ${Z}^{\mu }=\mathrm{span}\left\{{\epsilon }_{i}^{\mu }\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}1\le i\le {i}_{\mu }\right\}$ is isomorphic to the simple $\underset{\lambda }{\oplus }{M}_{{m}_{\lambda }}\left(\stackrel{‾}{𝔽}\right)$-module of column vectors of length ${m}_{\mu }$. The decomposition of $M$ as $A\oplus Z$-modules follows since $a ⊗ φ i j λ m ⊗ ε k μ = δ λ μ δ i j a ⊗ ε i μ , for all m ∈ A μ , a ∈ A .$ $\square$

## Reference

[HA] T. Halverson and A. Ram, Partition algebras, European Journal of Combinatorics 26, (2005), 869-921; arXiv:math/040131v2.