A Representation Theory Problem in Putnam Competition

Question

The following was the B6 problem of 1985 Putnam Competition: Suppose $G$ is a finite group (under matrix multiplication) of real $n\times n$ matrices $\{M_i\}, 1\leq i\leq r$. Suppose that $$\sum_{i=1}^r tr (M_i)=0$$, prove that $$\sum_{i=1}^rM_i=0.$$

Here is an official proof from the committee which I didn't understand:

Lemma: Let $G$ be a finite group of order $r$. Let $\rho: G\rightarrow GL(V)$ be a representation of $G$ on some finite dimensional vector space $V$. Then $$\sum_{g\in G}tr \rho_g$$ is a non-negative integer divisible by $r$, and is zero iff $$\sum_{g\in G}\rho_g=0$$.

Proof: Let $\chi_1,\cdots, \chi_s$ be the irreducible characters of $G$ and $\chi= \sum_{i=1}^s a_i\chi_i$ and $\psi=\sum_{i=1}^sb_i\chi_i$ be arbitrary characters. Then by the orthogonality relations of characters, we have $$\frac{1}{|G|}\sum_{g\in G}\chi(g)\overline{\psi (g)}=\sum_{i=1}^sa_ib_i$$. Applying this to the character of $\rho$ and the trivial character $\mathbb{1}$ shows that $\frac{1}{|G|} \sum_{g\in G}tr \rho_g$ equals the multiplicity of $\mathbb{1}$ in $\rho$, which is a non-negative integer. Now suppose that the matrix $S=\sum_{g\in G}\rho_g$ is non-zero. Choose $v\in V$ with $Sv\not=0$. The relation $\rho_hS=S$ shows that $Sv$ is fixed by $\rho_h$ for all $h\in G$. In other words, $Sv$ spans a trivial subrepresentation of $\rho$, so the non-negative integer of the previous paragraph is positive. QED

We now return to the problem at hand. "Unfortunately the $M_i$ do not necessarily define a representation of $G$, since the $M_i$ need not be invertible." Instead we need to apply the lemma to the action of $G$ on $\mathbb{C}^n/K$ for some subspace $K$ ...

I do not understand wthe sentence in "". Isn't the set of $M_i$'s form a group under multiplication? why they need not to be invertible? The above proof is copied from Kedlaya, Poonen and Vakil's Putnam competition 1985-2000. Thanks for helping

Answer 1

It seems to me that the proposers of this problem went an extra mile to be misunderstood: Most reasonable mathematicians would read "group under matrix multiplication" as "group whose multiplication is matrix multiplication and whose identity is the identity of matrix multiplication". Under this interpretation, the $M_i$ do define a representation of $G$. But apparently the problem did not mean to require the identity of the group to be the identity of matrix multiplication, and so you do not know if the matrices are actually invertible as matrices.

But ambiguity is not a one-player game. Consider $r = 3$, $M_1 = I_n$, $M_2 = \operatorname{diag}\left(1, -1\right)$ and $M_3 = \operatorname{diag}\left(1, -1\right)$. Oh, $M_1, M_2, \ldots, M_r$ are supposed to be distinct? Good to know. I am wondering how often this came up on appeal?

Anyway the solution you quoted is overkill. The problem straightforwardly generalizes to matrices over any field of characteristic $0$ instead of real matrices; good luck defining Hermitian forms over arbitrary fields. A solution that generalizes (and is a lot shorter and more elementary than the one in the original post) proceeds as follows (very roughly sketched):

We have \begin{align} \left(M_1 + M_2 + \cdots + M_r\right)^2 = \sum_{i, j} M_i M_j = \sum_{k} \sum_{\substack{i, j ;\\ \ M_i M_j = M_k}} M_k \end{align} (since the $M_i$ form a group, so each $M_i M_j$ equals some $M_k$), where all indices in sums range over $\left\{1,2,\ldots,r\right\}$. Now, for every $1 \leq k \leq r$, there exist precisely $r$ pairs $\left(i, j\right)$ such that $M_k = M_i M_j$ (again since the $M_i$ form a group). Hence, for every $1 \leq k \leq r$, we have \begin{align} \sum_{\substack{i, j ;\\ \ M_i M_j = M_k}} M_k = r M_k . \end{align} Thus, \begin{align} \left(M_1 + M_2 + \cdots + M_r\right)^2 &= \sum_{k} \underbrace{\sum_{\substack{i, j ;\\ \ M_i M_j = M_k}} M_k}_{=r M_k} = \sum_{k} r M_k \\ &= r\left(M_1 + M_2 + \cdots + M_r\right) . \end{align} This readily yields that $\dfrac{1}{r}\left(M_1 + M_2 + \cdots + M_r\right)$ is an idempotent. But the trace of an idempotent matrix equals its rank (this is a well-known fact), and this particular idempotent matrix $\dfrac{1}{r}\left(M_1 + M_2 + \cdots + M_r\right)$ has trace $0$. How many matrices with rank $0$ are there?