Let $\{A_i\}$ be a collection of $m$ hyperplanes in $\mathbb{C}^n$ which all pass through the origin (a **central hyperplane arrangement**). Such an arrangement is called **Coxeter** if reflecting across any hyperplane in $\{A_i\}$ sends the arrangement to itself (and so the reflections automatically will generate a Coxeter group).

Now, I will define a rather random-seeming condition on an arbitrary central arrangement. Choose a normal vector $n_i$ to each $A_i$. Consider the function on $\gamma$ on $\mathbb{C}^n$ given by
$$ \gamma(v) = \sum_{1\leq i< j\leq m}(n_i,n_j)\left(\prod_{k\neq i,j} (n_k,v) \right) $$
The function $\gamma$ depends on the specific choice of normal vectors; however, whether $\gamma$ vanishes does not depend on the choice of normal vectors (since scaling a normal vector will scale the output). Call a central hyperplane arrangement **puzzling** if $\gamma(v)=0$ for all $v$.

The 'puzzling' condition came up in studying a very specific research problem. However, both the context and a day's worth of experimentation have lead to the following conjecture.

**Conjecture:** The puzzling arrangements are exactly the Coxeter arrangements.

It's worth noting that I can't show either direction of the conjecture. Brute force computation says that for $n=2$, the conjecture is true.

Just from the form, it kind of reminds me of a Weyl character formula-type identity, but I don't really know much about those. My hope is that this kind of identity is pretty well known to people who work with such things.

**Edit:** There's another way of stating this identity, that's closer to the context in which I encountered it (differential operators). Let $n_i^*$ denote the function $(n_i,-)$, and let $d_i$ denote differentiation along the vector $n_i$. Then

$$ \gamma = \left( \sum_i (n_i^*)^{-2}(n_i^*d_i-(n_i,n_i))\right) \prod_j n_j $$

Therefore, the 'puzzling' condition is then a statement about the function $\prod_jn_j$ being killed by a particular rational differential operator.

**Edit 2:** Fixed the definition of Coxeter arrangements... I want all the reflections, not just a generating set.