Schur's lemma

LEMMA 7.7   Let $k$ be an algebraically closed field. Let $V$ be a finite dimensional representation of a $k$ -algebra $A$ . That means, $V$ is a finite dimensional vector space over $k$ , and we have a $k$ -algebra homomorphism

$$\alpha: A\to \operatorname{End}_k(V).$$

$V$ may be regarded as an $A$ -module. We assume further that $V$ is irreducible representation of $A$ . That means, $V$ admits no non trivial $A$ -module.

Then for each element $z$ the center $Z(A)$ of $A$ , $\alpha(z)$ is equal to a constant.

PROOF.. Let $c \in k$ be an eigen value of $\alpha(z)$ . Then $\alpha(z-c)$ has a non trivial kernel. That means,

$$V_c=\{ v\in V; \alpha(z-c) . v=0\}$$

is a non zero vector subspace of $V$ . It is easy to verify that $V_c$ is a $A$ -submodule of $V$ . From the irreducibility assumption, we have

$$V=V_c,$$

which in turn means that $\alpha(z)=c$ .

$\qedsymbol$

COROLLARY 7.8   Let $k$ be an algebraically closed field of characteristic $p\neq 0$ . Then every finite dimensional irreducible representation $\alpha:A_n(k)\to \operatorname{End}_k(V)$ of $A_n(k)$ is equivalent to a representation $\Phi_c$ for some $c\in k^{2 n}$ .

PROOF.. It is easy to see that $\gamma_j^p$ is in the center $Z(A_n(k))$ of the Weyl algebra $A_n(k)$ for $j=1,2,3,\dots,2 n$ .

From the Lemma above, we have

$$\alpha(\gamma_j^p)=a_j$$

for some $a_j\in k$ . Let $c_j$ be the $p$ -th root of $a_j$ in $k$ (which exists uniquely). Then we see that

$$\mathfrak{a}_c\subset \operatorname{Ker}(\alpha).$$

Thus $\alpha$ is essentially a representation of $A_n(k)/\mathfrak{a}_c\cong M_{p^n}(k)$ . $\qedsymbol$

For completeness's sake, we record here the following easy lemma.

LEMMA 7.9   Let $k$ be a field. Then any finite dimensional representation $M_n(k)$ is written as a direct sum of copies of the standard representation $k^n$ .

PROOF.. Let us denote by $e_{i j}$ the $i,j$ -elementary matrix. That means,

$$(e_{i j})_{k l}= \begin{cases} 1 & \text{ when } (i,j)=(k,l)\\ 0 & \text{otherwise.} \end{cases}$$

Let $V$ be the representation vector space. We first note that $p_i=e_{ii}$ form a complete system of mutually orthogonal projections. That means,

$$p_i^2=p_i, \qquad p_i p_j=0\quad ($$if $$i\neq j),\qquad \sum_{i=1}^n p_i=1$$

Let us thus put $V_i=p_i V$ . Then we have

$$V=\bigoplus_{i=1}^n V_n.$$

Furthermore,

$$e_{ij}. :V_j \to V_i$$

is an isomorphism of vector space whose inverse is equal to $e_{ji}$ .

Let us take a linear basis $\{v_l\}_{l=1}^d$ of $V_1$ over $k$ . Then

$$\{e_{i 1} v_l; i=1,2,3,\dots,n, l=1,2,3,\dots,d\}$$

is a basis of $V$ . It is now easy to see that for each $l$ , the vector space

$$W_l=$$linear span$$(\{e_{i 1} v_l; i=1,2,3,\dots,n\})$$

is isomorphic to the standard representation of $M_n(k)$ .

$\qedsymbol$

We also notice the following

COROLLARY 7.10   Let $k$ be a field of characteristic $p\neq 0$ . Then $M_{p^n}(k)$ is generated by $\{\mu_i\}_{i=1}^{2 n}$ such that

$$[\mu_i \mu_j]=h_{ij} \quad (\forall i,j ),\quad \mu_i^p=0 (\forall i)$$

PROOF.. We take the representation $\Phi_0$ above and

$$\mu_i=\Phi_0(\gamma_i) \qquad (i=1,2,3,\dots,2 n).$$

$\qedsymbol$