SLC 93 - Lecture 1: Nonnesting

### <u>How to read these slides in 3 easy$^*$ steps</u>

1. <a href="mailto:nathan.f.williams@gmail.com">Invite the author</a> to come present these slides to you.
2.
     (a) Read <a href="https://bookstore.ams.org/memo-202-949">Drew's book</a>.<br>
     (b) Read <a href="https://bookstore.ams.org/view?ProductCode=MEMO/305/1535">my book</a> (with Christian and Hugh)<br>
     (c) Read each paper in the <a href="./papers">short list of ~100 references</a> that I have helpfully compiled in chronological order for you. <font size=2> (Note: this list is not comprehensive!)</font><br>
     (d) Publish a paper on what you have learned from steps 2 (a)-(c).
3. Make sure you have several other experts on hand when the invited author does not arrive to present the slides to you.
 
<br><br><br><br>

<font size=2>
$^*$ OK, hard.  By the way, if you have any recollections (or corrections) for these slides, please <a href="mailto:nathan.f.williams@gmail.com">email me</a>.</font>

<u>A word from Vic **REINER**</u>

> <div style="font-size: .6em;"> I'll repeat here something that I think you already heard me say at LaCIM in <a href="https://www-users.cse.umn.edu/~reiner/Talks/CRM-LaCIM-SpringSchool-part1.pdf">this talk series</a>. The first three of the four formulas on page 3 of those slides, and their generalizations to reflection groups on page 8, were all things that I had heard about or read about at some point in grad school.

> <div style="font-size: .6em;">  I found them incredibly cool, because I loved it when type $A$ combinatorics turned out to generalize to reflection groups.  However, even though the Catalan numbers were ubiquitous in my grad school diet, that 4th formula generalizing to W-Catalan numbers really blew me over when I heard about it much later.  Even cooler for me was the fact that it related to that <a href="./papers/1996 - Edelman, Reiner - Free arrangements and rhombic tilings.pdf">conjecture on Catalan arrangements</a> that Edelman and I had made in our paper on rhombic tilings
(discussed in <a href="./papers/1997 - Reiner - Non-crossing partitions for classical reflection groups.pdf">Remark 2 of my $NC^B(n)$ paper</a>).</div>

> <div style="font-size: .6em;"> I was simiarly bowled over when I learned of Bessis's generalization of the Cayley-Hurwitz tree count $n^{n-2}$ to the formula $\prod_{i} \frac{ih}{d_i},$ as the degree of LL-covering. </div>

### <u> Picturesque parking</u> <font size=2><a href="./papers/1966 - Konheim, Weiss - An Occupancy Discipline and Applications.pdf">after Konheim and Weiss</a></font>

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<u>A word from Ira **GESSEL**</u>
> <div style="font-size: .6em;"> As I recall, I first learned about parking functions in the fall of 1977, when I had a one-year postdoctoral fellowship at the IBM Watson Research Center. I think maybe someone there told me about them, but I don't remember any details. I was at UCSD in the spring of 1978 and I told Adriano Garsia about them, so I may have been responsible for introducing them to the enumerative combinatorics community, though of course I can't know that for sure.  I don't remember when I talked to Mark Haiman about them, but I knew about the $(n+1)^{n-1}$ diagonal harmonics conjecture, and so parking functions were a natural thing to look at in trying to find an interpretation for $(n+1)^{n-1}$.
>$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Ira Gessel
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### <u> Parking</u>  <font size=2>(after [Pollak](./img/Pollak))</font>
 Pollak's *gorgeous* proof has appeared *many* times in the literature.  <br> Never by Pollak.
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### <u> Selected References</u> 
<center>
<a href="./papers/1994 - Haiman - Conjectures on the Quotient Ring by Diagonal Invariants.pdf"><img data-src="./img/Haiman - Conjectures Title.png" height="300"></a><a href="./papers/1996 - Stanley - Parking functions and noncrossing partitions.pdf"><img data-src="./img/Stanley - Chains Title.png" height="300"></a>
	<a href="./papers/2002 - Anderson - Partitions which are simultaneously t1- and t2-core.pdf"><img data-src="./img/Anderson - Title.png" height="300"></a>
<a href="./papers/2013 - Armstrong - Hyperplane arrangements and diagonal harmonics.pdf"><img data-src="./img/Armstrong - HADH.png" height="300"></a></center><center>
<a href="./papers/2016 - Gorsky, Mazin, Vazirani - Affine permutations and rational slope parking functions.pdf"><img data-src="./img/GMV - Title.png" height="300"></a>
<a href="./papers/2016 - Armstrong, Loehr, Warrington - Rational Parking Functions and Catalan Numbers.pdf"><img data-src="./img/ALW - Rational Title.png" height="300"></a>
<a href="./papers/2018 - Bodnar - Rational Catalan Combinatorics.pdf"><img data-src="./img/Bodnar - Title.png" height="300"></a>
<a href="./papers/2024 - McCammond, Thomas, Williams - Fixed Points of Parking Functions.pdf"><img data-src="./img/MTW - Title.png" height="300"></a></center>

### <u>The Shi arrangement and the Sommers region</u>

> **Def.** (Jian-Yi Shi). The <span style="color:#ff2626">*Shi arrangement*</span> of a root system $\Phi$ is
	$$\\{ H_{\alpha,0},  H_{\alpha,1}\\}_{\alpha \in \Phi^+}\subseteq \mathbb{R}^r.$$

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### <u>A word from Igor **PAK**</u>

> <div style="font-size: .6em;"> My involvement with Catalan numbers was very minor and fairly accidental.  I was never a major fan of the subject since I couldn't imagine there were new things to do there (are there still?). <br><br> However, I was always curious about why some combinatorial objects have a nice bijection between them and why others do not.  What would it mean to have a canonical bijection between the objects?  It's one of those questions which helps you write a large number of papers without ever getting close to the answer. <br><br> Anyway, at some point about 10 years ago I was wondering why there are so many different bijections between the same Catalan objects.  For example, at the time there were nine (!) bijections between 123 and 213-avoiding permutations.  Shouldn't that mean that none of them is canonical?  Eventually we worked out some sort of asymptotic answer to this question in a <a href="https://www.math.ucla.edu/~pak/papers/PermShapeLong.pdf">paper</a> with my former PhD student Sam Miner.  <br><br>
> While preparing the references to our paper, I realized that there are no web sources where the literature is organized and accessible, so I made a <a href="https://www.math.ucla.edu/~pak/lectures/Cat/pakcat.htm">Catalan Numbers Page</a>.  While making this website, I realized there are no good accounts of the history of Catalan numbers, so I added historical sources to the website, read them all, and wrote a <a href="https://igorpak.wordpress.com/2013/02/20/who-computed-catalan-numbers/">blogpost</a>.  While studying the history, it occurred to me that Catalan's involvement was relatively minor and it was really Euler, Goldbach and Segner who made the early key contributions.  So I used <a href="https://books.google.com/ngrams/">Google tools</a> to figure out who came up with the name.  It turned out to be Riordan who wanted to popularize the subject, and I am guessing liked the way the word "Catalan" looks and sounds.  I wrote it all up in another short <a href="https://igorpak.wordpress.com/2014/02/05/who-named-catalan-numbers/">blogpost</a>.  Stanley liked the story and asked me to write it up in <a href="https://www.math.ucla.edu/~pak/papers/cathist4.pdf">this appendix</a> to his <em>Catalan Numbers</em> book. <br><br> 
>$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Igor Pak</div>

### <u>The Shi arrangement and the Sommers region</u>

> **Thm.** (Jian-Yi Shi) Associate to a region of the Shi arrangement a  <span style="color:#ff2626">*sign type*</span>: record if the region is separated from the identity by: <br> <center> ($-$) $H_\alpha = 0,~~~~$  ($+$) $H_\alpha = 1,~~~~$ or$~~~~$ ($\bigcirc$) neither.</center> <font size=2>(Can think of this as generalizing the Park-Stanley labeling.)</font>

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### <u>The Shi arrangement and the Sommers region</u>

> **Thm.** (Jian-Yi Shi) There are $(h+1)^r$ Shi regions.<br>
	> **Proof.** The inverse of the affine Weyl group elements indexing the minimal alcoves in each region is an $(h+1)$-fold dilation of the fundamental alcove (called the <span style="color:#ff2626">*Sommers region*</span>).

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<u>A word from Eric **SOMMERS**</u>
> <div style="font-size: .5em; width:text-width; height:300px;overflow:auto">
I think the first time I heard about the region being named for me was when I was working at NSF and another program officer found a reference to it in a proposal.   <b>And I was like:  "What's the Sommers region?".</b> <br><br>   Later, Vazirani sent me her paper with Gorsky and Mazin where they define it.   And then there's your very nice paper "<a href="./papers/2024 - McCammond, Thomas, Williams - Fixed Points of Parking Functions.pdf">Fixed points of parking functions</a>" with McCammond and Thomas.<br><br>
The region is already used by Shi for $t = h+1$ where he counts the number of regions in his eponymous arrangement, by taking the inverse of the minimal affine Weyl group element in each region, so I certainly knew about that.  Then my mathematical brother, Ken Fan, had a paper "<a href="./papers/1996 - Fan - Euler characteristic of certain affine flag varieties.pdf">Euler characteristic of certain affine flag varieties</a>".  The region is in there:  Fan used it to parameterize the fixed points of an affine Springer fiber under a $\mathbb C^*$-action.   So I probably do not deserve credit for it.  It was in trying to extend Fan's result to parabolic versions of this kind of  affine Springer fiber that I got into this subject.  Then later, after the work of <a href="./papers/2002 - Cellini, Papi - Ad-nilpotent ideals of a Borel subalgebra II.pdf">Cellini and Papi</a>, the dominant regions turned out to be parametrized by the lattice points in the region.  <br><br>
Here is an old piece of scrap paper from around 1995 when I was a graduate student, trying to understand the affine paving of this affine Springer fiber (which known to have such a paving by <a href="./papers/1991 - Lusztig, Smelt - Fixed point varieties on the space of lattices.pdf">Lusztig-Smelt</a> in type A and then later by <a href="./papers/1997 - Goresky, Kottwitz, MacPherson - Equivariant cohomology, Koszul duality, and the localization theorem.pdf">Goresky, Kottwitz and MacPherson</a> in general type).    The cases of $t = 4, 5$ and $7$ are visible with the dimension of the affine cell that each chamber indexes.   So the complex affine cells give a Poincare polynomial is $1+3q+6q^2+6q^3$ for $t=4$, which is a well-known  single-grading of the parking functions for $S_3$ (those are $1$'s near the $0$).<br><br>
Recently my colleague, Alexei Oblomkov, gave two beautiful talks on his paper with Carlsson ("<a href="./papers/2025 - Carlsson, Oblomkov - Affine Schubert calculus and double coinvariants.pdf">Affine Schubert calculus and double coinvariants</a>").  You mentioned this paper in your article with McCammond and Thomas, so it's been around, but they recently updated it to give a second paving of the affine Springer fiber (at least in type A).  Here, the paving is by things that have the same Poincare polynomial as either 1) smooth (finite) Schubert varieties, or what is the same,  2) regular nilpotent Hessenberg varieties.   Each paved thing is indexed by an element of $S_n$ (for the $t=n+1$ case).    So this is a story that still has legs. 
>$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Eric Sommers</div>

###	<u>A word from Christophe **HOHLWEG** and Philippe **NADEAU**</u>

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<center><a href=""><img data-src="./img/Small Roots.jpg" height="250"></a></center>

1993.  Brigitte Brink and Robert Howlett.  A finiteness property an an automatic structure for Coxeter groups.  https://doi.org/10.1007/BF01445101
		2024.  Matthew Dyer, Susanna Fishel, Christophe Hohlweg, and Alice Mark.  Shi arrangements and low elements in Coxeter groups.  https://doi.org/10.1112/plms.12624-->

### <u>Rational Sommers regions</u>

> - Can use $\widetilde{W}$ to move the Sommers region to $(h+1)A_0$.<br>
	> - Dominant elements $\simeq$ coroot lattice $\check{Q}$, so $\check{Q}/(h+1)\check{Q}$.
	> - Generalizes to the rational Sommers region ($\simeq pA_0$) and $\check{Q}/p\check{Q}$ using affine roots of height $p$ coprime to $h$.
	> - *But no Shi arrangement!*

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> For $J \subseteq S$, let $W_J$ act on $\check{Q}/p\check{Q}$ for $p$ coprime to $h$.<br>
> **Note.** This is still a *permutation action*: 	$\chi_p(w) = p^{\mathrm{dim}(V^w)}$.<br><br>
> **Thm.** The number of $W_J$-orbits is $\mathrm{Cat}\_{p,J}(W) = \prod_{i=1}^r \frac{p+e_{i}^J}{d_{i}^J},$ where $e_i^J, d_i^J$ are the exponents/degrees of $W_J$ on $V$.  Specializations:
>$$\mathrm{Cat}\_{p,\emptyset}(W) = \prod_{i=1}^r \frac{p+0}{1} = p^r \text{ and } \mathrm{Cat}\_{p,S}(W) = \prod_{i=1}^r \frac{p+e_i}{d_i}.$$

<center>
<a href="./papers/1994 - Haiman - Conjectures on the Quotient Ring by Diagonal Invariants.pdf"><img data-src="./img/Haiman - Conjectures Title.png" height="300"></a>
<a href="./papers/1998 - Suter - The number of lattice points in alcoves and the exponents of the finite Weyl groups.pdf"><img data-src="./img/Suter - Title.png" height="300"></a>
<a href="./papers/2003 - Sommers - B-stable ideals in the nilradical of a Borel subalgebra.pdf"><img data-src="./img/bstable.png" height="300"></a>
<a href="./papers/2011 - Bessis, Reiner - Cyclic sieving of noncrossing partitions for complex reflection groups.pdf"><img data-src="./img/Bessis, Reiner - Title.png" height="300"></a>
</center>
<!--1998. Ruedi Suter. The Number of lattice Points in Alcoves and the Exponents of the Finite Weyl Groups.  https://doi.org/10.1090/S0025-5718-98-00919-3

1994. Mark Haiman. Conjectures on the Quotient Ring by Diagonal Invariants. https://doi.org/10.1023/A:1022450120589

Bessis Reiner - Cyclic sieving-->

### <u> Selected References</u>

### <u> Selected References</u>

</center>
<!--	1997. Jian-Yi Shi.  The number of \oplus-sign types.  https://doi.org/10.1093/qmath/48.1.93
	1997. Victor Reiner. Non-crossing partitions for classical reflection groups. https://doi.org/10.1016/S0012-365X(96)00365-2
	2004. Christos Athanasiadis and Victor Reiner.  Noncrossing Partitions for the Group $D_n$.  https://doi.org/10.1137/S0895480103432192
	2002. Paola Cellini and Paolo Papi. Ad-nilpotent ideals of a Borel subalgebra II. https://doi.org/10.1016/S0021-8693(02)00532-X
	2005.  Eric Sommers.  $B$-stable ideals in the nilradical of a Borel subalgebra.  https://doi.org/10.4153/CMB-2005-043-4
	2013. Drew Armstrong. Hyperplane arrangements and diagonal harmonics. https://doi.org/10.4310/JOC.2013.v4.n2.a2
	2015. Drew Armstrong, Victor Reiner, Brendon Rhoades.  Parking spaces.  https://doi.org/10.1016/j.aim.2014.10.012

2009. Drew Armstrong. Generalized noncrossing partitions and combinatorics of Coxeter groups.  https://doi.org/10.1090/S0065-9266-09-00565-1
	2005. Christos Athanasiadis. On a Refinement of the Generalized Catalan Numbers for Weyl Groups.  https://doi.org/10.1090/S0002-9947-04-03548-2
	2015. Michael Cuntz and Christian Stump. On root posets for noncrystallographic root systems.  https://doi.org/10.1090/s0025-5718-2014-02841-x-->

<u>A word from Minh-Tâm **TRINH**</u>
> <div style="font-size: .5em; width:text-width; height:250px;overflow:auto">
Fix a finite Weyl group $W$ with Coxeter number $h$. Fix also an integer $p > 0$ coprime to $h$ and a subset $J$ of the simple reflections.
>This data defines a complex projective algebraic variety with an (algebraic) action of $\mathbb{C}^\times$: an example of a parabolic affine Springer fiber. It is of a very specific kind, which is nowadays called "homogeneous of slope $p/h$".
The fixed points of the $\mathbb{C}^\times$-action are discrete. At the same time, projectivity means the underlying topological space of the affine Springer fiber is compact. So there must be finitely many fixed points.<br><br>
>The variety forms a subset of the partial affine flag variety $G(\mathbb{C}((z)))/K_J$ of a complex semisimple algebraic group $G$ with Weyl group $W$. When $W = S_n$, we can take $G = \mathrm{SL}_n$.
Here, $K_J$ is a "parahoric" subgroup, the affine/loop analogue of a parabolic subgroup of $G$, depending on $J$.<br><br>
The $\mathbb{C}^\times$-action on the affine Springer fiber is the restriction of a $\mathbb{C}^\times$-action on $G(\mathbb{C}((z)))/K_J$, whose fixed points are the cosets $\dot{w}K_J$ as $w$ runs over $W^{\mathrm{aff}, \mathrm{ext}}/W_J$.<br><br>
The initial data also defines a vector $(1/h) \rho_J^\vee \in V$.
Sommers's "<a href="./papers/1997 - Sommers - A family of affine Weyl group representations.pdf">Family of Affine Weyl Group Representations</a>" paper observes that the fixed points belonging to the affine Springer fiber are essentially indexed by the $w$'s such that ($(1/h) \rho_J^\vee) \cdot w^{-1}$ lands inside the closure of the $p$-dilated fundamental alcove, once we identify $V$ with the vector space spanned by the cocharacter lattice.
(Sommers speaks of the coroot lattice, but it is the same as the cocharacter lattice when $G$ is simply-connected.)<br><br>
When $J = S$, we find that $K_J = G(\mathbb{C}[[z]])$ and $W_J = W$ and $\rho_J^\vee$ is the zero vector. So the fixed points are indexed precisely by the coroots in the closed $p$-dilated fundamental alcove.
When $J = \emptyset$, we find that $K_J$ is an "Iwahori" subgroup of $G(\mathbb{C}((z)))$ and $W_J = \{e\}$ and $\rho_J^\vee$ is as generic as possible. Here the fixed points form a set of size $p^{r}$.<br><br>
The affine Springer fiber is a union of pairwise-disjoint strata, each forming an affine space of some dimension. We call this an affine paving.
The paving strata are in bijection with the $\mathbb{C}^\times$-fixed points. Given a fixed point, we look at all points $x$ in the affine Springer fiber such that $c\cdot x$ tends to that fixed point as $c \in \mathbb{C}^\times$ tends toward $0$: this is the corresponding stratum.
This situation ensures that the singular/analytic/Betti cohomology of the affine Springer fiber forms a vector space of the same dimension as the number of $\mathbb{C}^\times$-fixed points.
Usual Springer theory shows that there is a W-action on the cohomology. The character of this action, without any gradings, was computed by Sommers (his Thm. 7.3; more accurately, he takes the Euler characteristic of the cohomological degree, but the degrees occurring are all even so it doesn't matter).<br><br>
There are several candidates for a further filtration on the cohomology.
<a href="./papers/2014 - Hikita - Aﬃne Springer fibers of type A and combinatorics of diagonal coinvariants.pdf">Hikita</a> gave one specific to type $A$. He showed that the resulting bigraded $S_n$-module provides a "rational" ($p/h$) generalization of the space of diagonal harmonics, thus resolving various expectations of Haiman, Armstrong, etc.
Most of <a href="./papers/2014 - Hikita - Aﬃne Springer fibers of type A and combinatorics of diagonal coinvariants.pdf">Hikita's paper</a> works with $J = \emptyset$, since it is the richest case. But when we take $J = S$, the affine Springer fiber recovers the variety sometimes called the local compactified Jacobian of $y^h = x^p$, studied by e.g. Gorsky - Mazin.<br><br>
By work of <a href="./papers/1994 - Haiman - Conjectures on the Quotient Ring by Diagonal Invariants">Haiman</a>, <a href="./papers/2003 - Gordon - On the quotient ring by diagonal invariants">Gordon</a>, etc., this leads us to expect that in general type, the cohomology has an action of the rational Cherednik algebra of $W$ of parameter $p/h$.
Unfortunately, this doesn't seem to happen naturally outside of type $A$. Instead, in general type, the best we can do is build an RCA action on a modified vector space, where we take the $\mathbb{C}^\times$-equivariant cohomology and then specialize the equivariant parameter to a nonzero value. (Specializing to zero should recover ordinary cohomology.) To match the two constructions, we need a flatness result that no one knows how to prove.<br><br>
The RCA action on the modified cohomology is established in the first paper of <a href="./papers/2016 - Oblomkov, Yun - Geometric representations of graded and rational Cherednik algebras.pdf">Oblomkov - Yun</a>. Their second paper discusses the complications around the flatness result.
<a href="./papers/2016 - Oblomkov, Yun - Geometric representations of graded and rational Cherednik algebras">Oblomkov - Yun</a> do prove that their RCA module is the irreducible module we usually call $L(\mathrm{triv})$.
Any RCA module has a $W$-stable grading, coming from the action of a so-called "Euler element" that commutes with the subalgebra of the RCA formed by $\mathbb{C}[W]$.
For $L(\mathrm{triv})$, the resulting singly-graded $W$-module is precisely the Armstrong - Reiner - Rhoades parking space. The Euler grading arises from the so-called perverse filtration on the modified cohomology of the affine Springer fiber.<br><br>
The appendix by Etingof to <a href="./papers/2015 - Armstrong, Reiner, Rhoades - Parking spaces.pdf">[ARR]</a> explains how the RCA-module structure on $L(\mathrm{triv})$ is essentially equivalent to the existence of a hsop with certain properties in $\mathbb{C}[V]$, carrying $V^\ast$.<br><br>
>$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Minh-Tâm Trinh</div>

### <u> Algebraic parking spaces</u>
> **Caution.** For $W$ a well-generated complex reflection group, the quantity $\prod_{i=1}^r \frac{p+e_i}{d_i}$ is not usually an integer!

$$\text{On the other hand, }\mathrm{Cat}\_p(W):=\prod_{i=1}^r \frac{p+e_i(V^\sigma)}{d_i} \text{ is}\ldots$$

> **Ex.** For $I_2(5)$ and $p=2$, $\frac{(2+1)(2+4)}{2 \cdot 5}=\frac{9}{5}$ vs. $\frac{(2+2)(2+3)}{2 \cdot 5}=2.$

<!--	We want to have $W$-sets $S$ of size $p^r$ for $p$ coprime to $h$ as our ``rational parking functions'' such that $S^W = \prod_{i=1}^r \frac{p+e_i}{d_i}$, just like for Weyl groups.

But this isn't even usually an integer.  For example, for $I_2(5)$ with $h=5$ we would have for $p=2$ $(2+1)(2+4)/(2 \cdot 5)$.-->

### <u> Algebraic parking spaces</u>

> Let $W$ be a well-generated complex reflection group. <br>
	For any $p$ coprime to $h$, we can find a <span style="color:#ff2626">*homogenous system of parameters*</span> $\theta_1,\ldots,\theta_r$ of degree $p$ polynomials in $\mathbb{C}[V^*]$ such that:<br>
	> - $V \simeq \mathrm{span}\\{\theta_1,\ldots,\theta_r\\}$ <br>
	> - $\mathrm{Park}(W,p)=\mathbb{C}[V]/\mathbb{C}[\theta_1,\ldots,\theta_r]\_+$ is finite dimensional (dim=$p^r$) and<br>
	> - $\mathrm{dim}(\mathrm{Park}(W,p)^W) = \prod_{i=1}^r \frac{p+e_i(V^\sigma)}{d_i}$.<br>
	>
	> **Def.** (Armstrong, Reiner, Rhoades)  The <span style="color:#ff2626">*algebraic parking space*</span> is then defined to be $\mathrm{Park}(W,p)$.

> **Caution.** This is *not* a permutation representation in general:<br> 
	$\chi_p(w) = p^{\mathrm{dim}(V^w)} \prod \frac{1-\sigma{\lambda}}{1-\lambda}$ (product over all eigenvalues $\lambda \neq 1$ of $w$).

<u>A word from Stephen **GRIFFETH**</u>
> <div style="font-size: .5em; width:text-width; height:250px;overflow:auto">
[hsops] exist for well-generated complex reflection groups thanks to a combination of various results in the literature. Specifically, it follows from our paper with Iain Gordon "<a href="./papers/2012 - Gordon, Griffeth - Catalan numbers for complex reflection groups.pdf">Catalan numbers for complex reflection groups</a>," using Theorem 2.11 and the observations in subsection 2.18 about well-generated groups, together with the freeness theorem for finite Hecke algebras (see Etingof's paper <a href="./papers/2017 - Etingof - Proof of the Broué–Malle–Rouquier Conjecture.pdf">Proof of the Broué-Malle-Rouquier conjecture...</a>").<br><br>
The take-away messages are: <br>
> 1. The natural level of generality for things related to the diagonal coinvariant ring is that of quaternionic reflection 
groups. <br>
and<br>
2. The Coxeter number $h$ should be replaced by the number $g=2N/n$, where $N$ is the number of reflections in the irreducible quaternionic reflection group $W$ and $n$ is its rank. <br><br>
This is true even for the well-generated complex reflection groups $G(\ell,1,n)$, <center>where $h=n\ell$      but      $g=(n+1) \ell-2$.</center><br>
It turns out that for <em>all</em> irreducible complex reflection groups, there is a quotient of the diagonal coinvariant ring of dimension $(g+1)^r$ (so the 
quotient Iain and I were looking at is slightly wrong if one cares about the diagonal coinvariant ring); this is the main theorem of my paper 
"The diagonal coinvariant ring of a complex reflection group." For the quaternionic case, which is still conjectural in the exceptional cases, see 
my preprint with Cartaya "<a href="./papers/2024 - Cartaya, Griffeth - Zero Fibers of Quaternionic Quotient Singularities.pdf">Zero-fibers of quaternionic quotient singularities</a>." In the appendix there is a case-free proof that $g$ is an integer. I have 
not begun to think about the "Fuss" cases.<br><br>
At this moment, I do not know of a combinatorial definition that leads to the number $(g+1)^r$ that seems to be relevant in representation theory. 
There is some reason to be pessimistic in the quaternionic case: for the binary icosahedral group $W$, I do not believe that the zero-fiber ring carries a 
permutation representation of $W$ (even up to twisting by linear characters, since there aren't any non-trivial ones). So whatever combinatorics one 
might invent or discover, it won't be as straightforward as finding a $W$-set with the right cardinality. 
>$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Stephen Griffeth
</div>
<center><a href="./papers/2012 - Gordon, Griffeth - Catalan numbers for complex reflection groups.pdf"><img data-src="./img/Gordon, Griffeth - Title.png" height="300"></a>$~~~~~$<a href="./papers/2017 - Etingof - Proof of the Broué–Malle–Rouquier Conjecture.pdf"><img data-src="./img/Etingof - Title.png" height="300"></a>$~~~~~$<a href="./papers/2022 - Griffeth - The diagonal coinvariant algebra of a complex reflection group.pdf"><img data-src="./img/Griffeth - Title.png" height="300"></a>$~~~~~$<a href="./papers/2024 - Cartaya, Griffeth - Zero Fibers of Quaternionic Quotient Singularities.pdf"><img data-src="./img/Cartaya, Griffeth - Title.png" height="300"></a></center>
<!--Proof of the Broué-Malle-Rouquier conjecture

The diagonal coinvariant ring of a complex reflection group.

Zero-fibers of quaternionic quotient singularities-->

### <u> Selected References</u>

<center>
<a href="./papers/2012 - Gordon, Griffeth - Catalan numbers for complex reflection groups.pdf"><img data-src="./img/Gordon, Griffeth - Title.png" height="400"></a>
<a href="./papers/2015 - Armstrong, Reiner, Rhoades - Parking spaces"><img data-src="./img/ARR - Title.png" height="400"></a>
<a href="./papers/1994 - Haiman - Conjectures on the Quotient Ring by Diagonal Invariants.pdf"><img data-src="./img/Haiman - Conjectures Title.png" height="400"></a>
</center>
<!--	1997. Jian-Yi Shi.  The number of \oplus-sign types.  https://doi.org/10.1093/qmath/48.1.93
	1997. Victor Reiner. Non-crossing partitions for classical reflection groups. https://doi.org/10.1016/S0012-365X(96)00365-2
	2004. Christos Athanasiadis and Victor Reiner.  Noncrossing Partitions for the Group $D_n$.  https://doi.org/10.1137/S0895480103432192
	2002. Paola Cellini and Paolo Papi. Ad-nilpotent ideals of a Borel subalgebra II. https://doi.org/10.1016/S0021-8693(02)00532-X
	2005.  Eric Sommers.  $B$-stable ideals in the nilradical of a Borel subalgebra.  https://doi.org/10.4153/CMB-2005-043-4
	2013. Drew Armstrong. Hyperplane arrangements and diagonal harmonics. https://doi.org/10.4310/JOC.2013.v4.n2.a2
	2015. Drew Armstrong, Victor Reiner, Brendon Rhoades.  Parking spaces.  https://doi.org/10.1016/j.aim.2014.10.012