## The short version

Here is an extremely natural hyperplane arrangement in $\mathbb{R}^n$, which I will call $R_n$ for *resonance arrangement*.

Let $x_i$ be the standard coordinates on $\mathbb{R}^n$. For each nonempty $I\subseteq [n]=\{1,\dots,n\}$, define the hyperplane $H_I$ to be the hyperplane given by $$\sum_{i\in I} x_i=0.$$ The resonance arrangement is given by all $2^n-1$ hyperplanes $H_I$. The arrangement $R_n$ is natural enough that it arises in many contexts -- to first order, my question is simply: have you come across it yourself?

This feels rather vague to be a good question, and after giving some background on where I've seen this I'll try to be a bit more specific about what I'm looking for, but my point is this arrangement has a rather simple and natural definition, and so crops up in multiple places, and I'd be curious to hear about more of them even if you can't specifically connect it to what follows.

## What I knew until this week

I came across this arrangement in my study of double Hurwitz numbers -- they are piecewise polynomial, and the chambers of the resonance arrangement are the chambers of polynomiality. I don't want to go into this much more, as it's unimportant to most of what follows Though I will say that conjecturally double Hurwitz numbers could be related to compactified Picard varieties in a way which would connect this arrangement up with birational transformations of those. Also, the name "resonance arrangement" was essentially introduce in this context, by Shadrin, Shapiro and Vainshtein.

It apparently comes up in physics -- I only know this because the number of regions of $R_n$, starting at $n=2$, is 2, 6, 32, 370, 11292, 1066044, 347326352 ... Sloane sequence A034997, which you will see was entered as "Number of Generalized Retarded Functions in Quantum Field Theory" by a physicist.

You might expect by that rate of growth that this hyperplane arrangement is completely intractable, and more specifically, what I would love from an answer is some kind qualitative statement about how ugly the $R_n$ get. Which brings us to:

## Connection to the GGMS decomposition

I got to thinking about this again now and decided to post on MO of it because of Noah's question about the vertices of a variation of GIT problem, where this arrangement is lurking around -- see there for more detail. Allen's brief comment there prompted me to skim some of his and related papers to that general area, and I found the introduction to Positroid varieties I: juggling and geometry most enlightening, together with the discussion at Noah's question could give another suggestion why $R_n$ is perhaps intractable. Briefly:

The arrangement $R_n$ is a natural extension of the $A_n$ arrangement. One common description of the $A_n$ arrangement is as the $\binom{n+1}{2}$ hyperplanes $y_i-y_j=0, i,j\in [n]$ and $y_i=0, i\in [n]$. However, one can consider the triangular change of variables $$y_k\mapsto \sum_{j\leq k} y_j$$.

This changes the hyperplanes to $$\sum_{i\leq k \leq j} y_k=0.$$

These hyperplanes are no longer invariant under permutation of the coordinates, and if we proceed to add the entire $S_n$ orbit of them, we get the resonance arrangement $R_n$.

From the discussion on Noah's question and the introduction to the Positroid paper, we see that this manipulation is a shadow of the GGMS decomposition of the Grassmannian, and describing this decomposition in general seems to be intractable in that it requires identifying whether matroids are representable or not. So, what I'd really like to know how is much of the "GGMS abyss", as they refer to it, is reflected in the resonance arrangement? Is it hopeless to describe and count its chambers?