# Generalization of Schur's lemma

I would like to proof a generalization of Schur's lemma for representations.

(Schur's lemma) (cfr. Jean-Pierre Serre, Linear representations of finite groups)

Let $\rho^1$: G $\to$ GL($V_1$) and $\rho^2$: G $\to$ GL($V_2$) be two irreducible representations of G, and let $f$ be a linear mapping of $V_1$ into $V_2$ such that $\rho^2_s \circ f = f \circ \rho^1_s$ for all $s \in$ G. Then:

(1) If $\rho^2$ and $\rho^2$ are not isomorphic, we have $f=0$.

(2) If $V_1 = V_2$ and $\rho^1 = \rho^2$, $f$ is a homothety (i.e., a scalar multiple of the identity).

Now I would like to make a generalization for reducible representations. Suppose that $\rho^1$: G $\to$ GL($V_1$) and $\rho^2$: G $\to$ GL($V_2$) are two representations (not necessarily irreducible) of G, and let f be a linear mapping of $V_1$ into $V_2$ such that $\rho_s^2 \circ f = f \circ \rho^1_s$ for all $s \in$ G. If we decompose $V_1$ and $V_2$ in irreducible representations:

$V_1= W_{1,1} \oplus W_{1,2} \oplus \cdots \oplus W_{1,k_1} \oplus W_{2,1} \oplus \cdots \cdots \oplus W_{m,1} \oplus \cdots \oplus W_{m,k_m}$

$V_2= W_{1,1} \oplus W_{1,2} \oplus \cdots \oplus W_{1,l_1} \oplus W_{2,1} \oplus \cdots \cdots \oplus W_{m,1} \oplus \cdots \oplus W_{m,l_m}$

where all the $W_{1,i}$ are isomorphic, all the $W_{2,i}$ are isomorphic, $\ldots$

Now, $f$ has the following block-matrix form.

$$\left[ \begin{matrix} & W_{1,1} & W_{1,2} & \cdots & W_{1,k_1} & W_{2,1} & \cdots & \cdots & W_{m,1} & \cdots & W_{m,k_m} \\ W_{1,1} & \lambda_{1,1} & \lambda_{1,2} & \cdots & \lambda_{1,k_1} & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ W_{1,2} & \lambda_{2,1} & \lambda_{2,2} & \cdots & \lambda_{2,k_1} & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ W_{1,l_1} & \lambda_{l_1,1} & \lambda_{l_1,2} & \cdots & \lambda_{l_1,k_1} & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ W_{2,1} & 0 & 0 & \cdots & 0 & \mu_{1,1} & \cdots & \cdots & 0 & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ W_{m,1} & 0 & 0 & \cdots & 0 & 0 & \cdots & \cdots & \omega_{1,1} & \cdots & \omega_{1,k_m}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ W_{m,l_m}& 0 & 0 & \cdots & 0 & 0 & \cdots & \cdots & \omega_{l_m,1} & \cdots & \omega_{l_m,k_m} \end{matrix} \right]$$

The first row and column are actually indices for the matrix (I didn't now how to do the typesetting). The greek letters are multiples of the identity matrix (it are homotheties).

All the notations seems very difficult but the result follows intuitively from Schur's lemma. I just don't know how to make this intuition rigorous. Clearly, our first step will be to restrict $f$ to an irreducible subrepresentation of $V_1$. But how do we move on?

As a consequence, can someone give a rigorous argument that the dimension of Hom($\rho^1$,$\rho^2$) is equal to the number of greek letters in this matrix.

You need to work over an algebraically closed field to state Schur's Lemma in the form given in Serre's book. Your generalization is basically correct (in that situation). It is probably easiest to break the proof into two cases : if $\rho^{1}$ and $\rho^{2}$ have no common irreducible constituent, then ${\rm Hom}(\rho^{1},\rho^{2}) = 0.$ And if $\rho^{1}$ is a a direct sum of $t$ isomorphic irreducible constituents, then ${\rm Hom}(\rho^{1},\rho^{1}) \cong M_{t}(F),$ where $F$ is the algebraically closed field you are working over.
• Can you add some more information about the notation $M_t(F)$. I'm used to Serre's approach rather than the approach using modules. A proof for $\mathbb{C}$ should be sufficient. – Michiel Van Couwenberghe Mar 9 '14 at 22:14
• That notation is intended to denote a full $t \times t$ matrix algebra over $F$ ( though in the isomorphism you exhibit, the copy of $F$ is seen via $d \times d$ scalar matrices, where $\rho^{1}$ has dimension $d.$ – Geoff Robinson Mar 9 '14 at 23:01