# Action of Weyl group on regions of Shi arrangement

This is an elaboration of a question which was aked on MO several years ago, which was unanswered but deleted by the question-asker. I hope it is okay to elaborate on a deleted question like this; for reference the question is at https://mathoverflow.net/questions/209856/ (probably only visible if you have enough reputation).

Here's the set-up. Let $$\Phi$$ be a crystallographic root system in vector space $$V$$ with Weyl group $$W$$. The Shi arrangement of $$\Phi$$ is the collection of hyperplanes $$\{H_{\alpha,k}\colon \alpha \in \Phi^+, k=0,1\}$$, where $$H_{\alpha,k} := \{v\in V\colon \langle v, \alpha \rangle = k\}$$. The number of regions of the Shi arrangement is known to be the number of "$$\Phi$$-parking functions", i.e., the number of elements of the finite torus $$Q/(h+1)Q$$, where $$Q := \mathbb{Z}\Phi$$ is the root lattice of $$\Phi$$ and $$h$$ the Coxeter number. See for instance the original paper of Shi Sign types corresponding to an affine Weyl group for a proof of this.

The question is about upgrading this numerical equality to an equality of $$W$$-actions.

Each region $$R$$ of the Shi arrangement is made up of a number of alcoves, where alcoves are the regions of the infinite hyperplane arrangement $$\{H_{\alpha,k}\colon \alpha \in \Phi^+, k\in\mathbb{Z}\}$$ corresponding to the affine Weyl group. I believe that each region contains a "minimal" alcove, which could be defined as the one closest to the origin (and probably could also be defined in terms of minimal length in the affine Weyl group). Let me use $$\mathrm{min}(R)$$ to denote this minimal region.

Correction: Originally I proposed an "action" here which is clearly not a group action. Let me reformulate the question in a way which has a hope of being correct.

Question: Is there some way to define an action of $$W$$ on the regions of the Shi arrangement so that:

• for a region $$R$$ and $$w\in W$$, we have $$w\cdot R=R'$$ if $$w\cdot\mathrm{min}(R)=\mathrm{min}(R')$$ for some region $$R'$$;
• the action isomorphic to the action of $$W$$ on $$Q/(h+1)Q$$?

Note that when $$w\cdot\mathrm{min}(R) \neq \mathrm{min}(R')$$ for any $$R'$$, it is not clear how to define $$w\cdot R$$.

The original question was about the Type A case. In fact, the asker proposed that the isomorphism should be given by the bijection of Athanasiadis and Linusson (see A Simple Bijection for the Regions of the Shi Arrangement of Hyperplanes) between regions of the Shi arrangement and parking functions.

Note that in their paper Parking Spaces, Armstrong, Reiner, and Rhoades explain, following Shi and Cellini and Papi, a canonical and type-uniform way to label the regions of the Shi arrangement by parking functions (see Section 10).

But if we compare the figure on page 42 of the Armstrong-Reiner-Rhoades paper to Figure 4 in the Athanasiadis-Linusson paper, we see the labelings are not the same in Type A, and in particular the Armstrong-Reiner-Rhoades is not compatible with the Shi arrangement action I described above.

I like this idea because of how geometric it is.

Here's a picture of what the orbits of this action should look like in Type $$A_2$$ (the red dot is the origin, and the dominant chamber is to the upper-right): Again, compare to Figure 4 of the Athanasiadis-Linusson paper.

Meanwhile, here's what I think it looks like in Type $$B_2$$ (ignore the labels, they are from the Armstrong-Reiner-Rhoades paper):

• Since the question was deleted, obviously the asker doesn't want it here under their name, so it may be appropriate to remove the identity of the asker. Perhaps a compromise would be that it is perfectly possible to link even to deleted questions, as mathoverflow.net/questions/209856 (which, for a non-moderator, just says that the question has been voluntarily removed by its author). This both removes the need separately to mention the question number, and allows the original asker to claim credit for the original question if desired. Sep 5, 2020 at 19:32
• @LSpice: good call. Done. Sep 5, 2020 at 19:37
• Ok, I haven't gone through all the details, but now I think the distinction between the Armstrong-Reiner-Rhoades labeling and the Athanasiadis-Linusson-style labeling is: in the former, they use ceilings of regions (hyperplanes $H_{\alpha,1}$ you cross moving in the dominant direction); for the latter, you should do the same thing but using floors (hyperplanes $H_{\alpha,1}$ you cross moving in the anti-dominant direction). Viewing the poset of dominant regions of the Shi arrangement as the root poset order ideal lattice $J(\Phi^+)$, this changes label $\max(I)$ to $\min(\Phi^+\setminus I)$. Sep 6, 2020 at 15:09

1. When you say "I believe that each region contains a "minimal" alcove, which could be defined as the one closest to the origin (and probably could also be defined in terms of minimal length in the affine Weyl group)" it is indeed the case, you can see it in the paper Sign types corresponding to an affine Weyl group of Shi that you cited. Shi gives actually two characterizations of this minimal element: Propositions 7.2 and 7.3.

2. For the way to define something that makes sense on the set of Shi regions you can use this. First of all notice that I take the notations of Shi. Let $$w \in W_a$$ and $$s \in S \subset W$$. We have the formula $$k(sw, \alpha) = k(w, s(\alpha)) + k(s, \alpha)$$. Since a Shi region is just the data of symbols $$\{-,0,+\}$$ over each position of $$\Phi^+$$, with some specific conditions (that Shi calls admissible sign types), you can try to see how the value of $$k(sw, \alpha)$$ behaves and then, maybe, by generalizing this formula for any $$x \in W$$, you might get something.