SLC 93 - Lecture 2: Noncrossing

### <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.

### <u> Factorizations of a Coxeter element</u>

>We can vary this problem in **many** ways.
>
> **Thm.** (Guillaume Chapuy and Christian Stump) Let $W$ be an irreducible well-generated complex reflection group of rank $r$ with Coxeter element $c$.  Then<br>
$$\mathrm{FAC}_W(t)=\sum\_{\ell \geq 0 } \frac{t^\ell}{\ell!} \\# \\{\text{len }\ell \text{ } R\text{-facts of } c\\}= \frac{1}{|W|}(e^{tN/r}-e^{-tN^*/r})^r$$

<u>A word from Jean **MICHEL**</u>
> <div style="font-size: .5em; width:text-width; height:250px;overflow:auto">
[This] story is how I came to work on the <a href="./papers/2014 - Stump, Chapuy - Counting factorizations of Coxeter elements into products of reflections">Chapuy-Stump</a> formula. Christian
explained  it to me when we met at a conference in Bielefeld. They prove it
by a character computation in every type of well-generated group. But where
the  proof is nicest is in type $A$ due  to the key feature in that type that
the  <a href="./2022 - Reeder - Weyl Group Characters Afforded By Zero Weight Spaces.pdf">only characters  which do  not vanish  on the  Coxeter element are the
exterior  powers of the reflection representation</a>. This had a similarity to
what  I knew of the cohomology  of the <a href="./papers/1976 - Deligne, Lusztig - Representations of Reductive Groups Over Finite Fields.pdf">Deligne-Lusztig Coxeter variety</a>: the
only  characters of the principal series which appear in the cohomology are
those   which  correspond  to   the  exterior  powers   of  the  reflection
representation of the Weyl group, and this in all types. I realized that if
one  could replace irreducible characters by almost characters the proof in
type $A$ would work for all Weyl groups. I managed to do it a few days later.
<br><br>
The  story for  the <a href="./papers/2022 - Chapuy, Douvropoulos - Counting chains in the noncrossing partition lattice via the W-Laplacian.pdf">Chapuy-Douvropoulos  paper</a> is  similar. Theo  sent me a
preliminary  version, and I noticed that a  lemma (what is Lemma 7.2 in the
published version) would imply that Lusztig's (truncated) Fourier transform
would  preserve  tower  equivalence.  The  rest  of my paper followed. But,
actually  the lemma in the  preliminary version was wrong:  it was much too
strong and just false. They had trouble fixing it, and it is much harder to
deduce  from  the  current  version  that Fourier transform preserves tower
equivalence!  I guess if I had just  the current version I would never have
noticed what I did...
>$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Jean Michel
</div>

<center><a href="./papers/2016 - Michel - Deligne-Lusztig theoretic derivation for Weyl groups of the number of reflection factorizations of a Coxeter element.pdf"><img data-src="./img/Michel 1.png" height="300"></a>$~~~~~$<a href="./papers/2022 - Chapuy, Douvropoulos - Counting chains in the noncrossing partition lattice via the W-Laplacian.pdf"><img data-src="./img/Theo 1.png" height="300"></a>$~~~~~$<a href="./papers/2023 - Michel - Tower equivalence and Lusztig’s truncated Fourier transform.pdf"><img data-src="./img/Michel 2.png" height="300"></a></center>

<u>A word from Theo **DOUVROPOULOS**</u>
> <div style="font-size: .5em; width:text-width; height:500px;overflow:auto">
For me, the big part of recent work on NC started with Theorem 4 of my paper with Chapuy (https://arxiv.org/pdf/2012.04519). Originally I had proved it to produce a forest version of our $W$-Matrix-Tree theorem but it was a very appealing formula and reminiscent to how I wanted to use on my thesis the Fomin-Reading recursion to match uniformly the degree of the LL map with a the chain number. As I was playing with this Theorem 4, I sort of accidentally realized that I could use it instead of the LL map (which was more difficult to understand) to prove a recursion on the formula $h^r r!/|W|$ that matched the recursion on the combinatorics of chains. This was a very happy moment as I was traveling in summer '19 to this conference in the <a href="https://www.icms.org.uk/workshops/2019/integrable-systems-special-functions-and-combinatorics">Isle of Skye</a> where I would give a talk about Frobenius manifolds that I really didn't know much about.<br><br>
So very quickly I did the proof of the (much later appearing) <a href="https://arxiv.org/abs/2109.04341">paper</a>. After that, I was trying to see if I could push this idea to the Catalan numbers case (or the whole equivariant structure). I tried essentially to realize the Fomin-Reading recursion on non-nesting objects (i.e. $(mh+1)A_0$) or just on the formulas. This was hard, but I tried various tricks including attempting to come up with a version of the Fomin-Reading recursion on the formulas side, but summing over arbitrary flats not just over maximal rank ones; in this way I guessed <a href="https://arxiv.org/pdf/2109.04341">formula (12)</a> which if proven would easily give uniform proofs of the $W$-isomorphism of the noncrossing and algebraic parking spaces. The combinatorial generalization of the Fomin-Reading recursion (on the generalized cluster complex side) was already done by Matthieu in our first paper.<br><br>
Then, at some point I realized that I could interpret some of these factors of formula 12 as free exponents on some multi-arrangements of the restrictions $\mathcal{A}^H$ on hyperplanes. I knew what I wanted the formulas to eventually look like and I was lucky because I had a good candidate for the multiplicity functions: in that same conference at the Isle of Skye I had met another greek mathematician, Giorgios Antoniou who was a student of Feigin and he told me about his <a href="https://arxiv.org/abs/2008.10133">thesis</a>: they were constructing a natural Coxeter-number-like statistic on the hyperplanes $Z$ in reflection restrictions $\mathcal{A}^X$ (this is still not well known). Putting these together I came up with Conjecture 1 of my <a href="https://drive.google.com/file/d/1H32gxONbG0jawyej8EvZbQA4q7MMBEDV/view">current research statement</a>. I tried to solve it with Stump and Feigin for more than a year during the pandemic and when we failed we instead did the conference in Edinburgh where you spoke.<br>
Eventually I proved half of that conjecture myself (this is the FPSAC 2025 talk). But regardless I was able to prove a recursion similar to that <a href="https://arxiv.org/pdf/2109.04341">formula (12)</a>  independently just extending the $W$-Laplacian ideas. This recursion was not as strong as (12) though and it took me 1-2 years between '21-'22 to complete the missing point in a different way. I finished the proof only in summer '22 and that was the FPSAC 2023 talk.<br><br>
You can interpret it a bunch of ways: counting chains in $NC(W)$ by the type of the $k$-th element; as Headley’s recursion on the Poincare polynomials of the Shi arrangements; as the local to global recursions on the free, constant multiplicity=2, multireflection arrangements; as a local-to-global expansion of the spectrum on the W-Laplacian; as the ramification formula for the LL map.<br>
>$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Theo Douvropoulos</div>

<u>A word from Frédéric **CHAPOTON**</u>
> <div style="font-size: .5em; width:text-width; height:250px;overflow:auto">
One tiny and maybe a bit strange thing that helped me to discover both the uniform formula for the Zeta polynomial of the noncrossing lattices and the uniform polynomial for the number of reflexions with full-support is the following idea.<br>
Irreducible Coxeter groups in rank 2 are parameterized by their Coxeter number $h$ which can be any integer greater than or equal to 2. This gives the series $I_2(h)$ of dihedral groups.<br><br>
It seems that many formulas for Coxeter groups of rank 3 can also be expressed in terms of the Coxeter number $h$ only. In particular, everything that depends only on the exponents of the Coxeter group can be expressed so. It turns out that this gives integers if and only if $h+2$ is a divisor of 24. So we get $h=4, 6, 10$ for $A_3, B_3, H_3$ and a mysterious non-existing Coxeter group of rank 3 with $h=22$. What is the meaning of this unorthodox $h=22$ case ? Why is the famous number 24 and its divisors appearing here ?
>$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Frédéric Chapoton
</div>

<center><a href="https://irma.math.unistra.fr/~chapoton/trinites.html"><img data-src="./img/Remarkable Trinities.png" height="300"></a>
<a href="./papers/2006 - Chapoton - Sur le nombre de réflexions pleines dans les groupes de Coxeter finis.pdf"><img data-src="./img/Chapoton - Reflection Title.png" height="300"></a>$~~~~~$</center>

### <u> Selected References</u> 
<center>
<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/2006 - Chapoton - Sur le nombre de réflexions pleines dans les groupes de Coxeter finis.pdf"><img data-src="./img/Chapoton - Reflection Title.png" height="300"></a>
<a href="./papers/1974 - Deligne - Letter to Looijenga.pdf"><img data-src="./img/Deligne Cover.png" height="300"></a>
<a href="./papers/2008 - Reading - Chains in the noncrossing partition lattice.pdf"><img data-src="./img/Reading chains Title.png" height="300"></a>

<u>A word from Christos **ATHANASIADIS**</u>
> <div style="font-size: .5em; width:text-width; height:250px;overflow:auto">
While at Penn, I realized the equienumeration of noncrossing and nonnesting partitions by block sizes. Perhaps the case of nonnesting partitions involved a 'finite field method' argument, for which I was looking for applications at that time. Then it seemed plausible to me that the same should hold for other types. The correct definition for noncrossing partitions of type $D$ was not available at that time.
<br><br>
During 2000-2001 I took a break to do my military service. After
that I was looking to reconnect to research, and Coxeter-Catalan
combinatorics seemed a promising direction, given the progress that had
happened in the meantime.<br><br>
First, I realized I could work out another application of the finite field method, which was left over from my thesis work and wrote <a href="./papers/2004 - Athanasiadis - Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes.pdf">that paper in Bull. London Math. Soc.</a> I was not expecting that there would be much interest about that result.<br><br>
Then I proved the new interpretation of type $D$ noncrossing partitions and showed
it to Vic when I visited him in Minneapolis. I asked him to work together on <a href="./papers/2004 - Athanasiadis, Reiner - Noncrossing Partitions for the Group D_n.pdf">the paper</a>, given his experience on the topic. I had a hard time to convince him that it was worth working on type $D$ noncrossing partitions. I even said that if we don't publish the new interpretation, sooner or later someone else will rediscover it. Indeed, a year later <a href="./papers/2006 - Bessis, Corran - Non-crossing partitions of type (e, e, r)">Bessis and Corran</a>, if I remember well, wrote to us that they had also come up with it (but they came after us).
<br><br>
Then I started thinking about the $m$-Catalan generalizations, given <a href="./papers/2004 - Athanasiadis - Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes.pdf">my paper in BLMS</a>, and wrote <a href="./papers/2005 - Athanasiadis - On a Refinement of the Generalized Catalan Numbers for Weyl Groups.pdf">the paper in TAMS</a> (all that while I was in Crete). Then I started thinking about shellability and attended that workshop in AIM. During Tom Brady's talk, I realized he had the part I was missing for the proof. Thus, we wrote <a href="2006 - Athanasiadis, Brady, McCammond, Watt - h-vectors of generalized associahedra and noncrossing partitions.pdf">the paper in Proc AMS.</a> I was sharing an office with Stembridge for a month in Mittag-Loeffler and submitted the paper to him.
<br><br>
At some point I started running out of ideas (and people close to me to talk) and thought I should start working on something else.
>$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Christos Athanasiadis
</div>
<center><a href="./papers/2004 - Athanasiadis, Reiner - Noncrossing Partitions for the Group D_n.pdf"><img data-src="./img/Athanasiadis, Reiner - Title.png" height="300"></a></center>

### <u> ...and the braid group</u>

> The poset of noncrossing partitions is the absolute order interval $[e,c]_R$, where $c$ is a Coxeter element of a well-generated complex reflection group $W$.
>
> **Thm.** (Birman-Ko-Lee, Brady-Watt, Bessis) The poset of noncrossing partitions is a lattice. It gives a "dual" presentation of the braid group: $$\pi_1(V^{\mathrm{reg}}/W)=B_W = \langle R : \text{set all words for }c\text{ equal}\rangle.$$

<u>A word from Joan **BIRMAN**</u>
> <div style="font-size: .5em; width:text-width; height:350px;overflow:auto">
Mathematics is filled with surprises about unexpected connections between different specialties, and the paper that I wrote with Ko and Lee, "A new approach to the word and conjugacy problem in braid groups", began in the 1990’s with a related surprise for me personally: an invitation to visit South Korea and give a series of lectures to grad students and visitors at a math institute called KAIST in the city of Taejon, on topics of my choice. I did not know my host, Professor Ki Hyoung Ko, and had never been to Korea. I got out a map, located Taejon (it's far from Seoul, the capital), discussed the trip with my husband, and since it turned out that he would be away the same two weeks and it sounded like an adventure, I accepted.<br><br>
It turned out to be a very busy and productive week. I met many people, had a great time (and worked very hard), and the food was delicious. Every night after dinner I worked in my hotel room to prepare what to say the next day.  Sang Jin Lee was one of Ko's graduate students at the time, and the paper I wrote with Ko and Lee resulted from discussions we had that week, which ultimately lead to a return visit by Ko to Columbia University in New York.<br><br>
Another VERY big surprise was when David Bessis proved that all of the groups that had already been dubbed 'Garside groups' by Patrick Dehornoy had not only two but even more <u>exactly</u> two Garside structures, generalizing and extending in major ways what we had proved to be the case for Artin's sequence of braid groups.<br><br>
A final surprise was when I learned, in 2025, about the far-reaching consequences of David Bessis' work.<br>
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Joan Birman
</div>
<center><a href="./papers/1998 - Birman, Ko, Lee - A new approach to the word and conjugacy problems in the braid groups.pdf"><img data-src="./img/Birman - Title.png" height="300"></a>$~~~~~$<a href="./papers/2002 - Brady, Watt - K(pi,1)'s for Artin groups of finite type.pdf"><img data-src="./img/Brady - Title.png" height="300"></a>$~~~~~$<a href="./papers/2003 - Bessis - The dual braid monoid.pdf"><img data-src="./img/Bessis - Title.png" height="300"></a>

<u>A word from Thomas **BRADY**</u>
> <div style="font-size: .5em; width:text-width; height:250px;overflow:auto">
Nevertheless, I have been reading the <a href="https://www.amazon.com/Mathematica-Secret-World-Intuition-Curiosity/dp/0300270887">wonderfully honest recent book of David Bessis</a> - you might recommend to your audience that they have their Math library get a copy - and am impressed by his description of the process of producing Mathematics.  <br><br>
In keeping with his view of the importance of a personal mental image and the slow process (in my case, positively glacial) of refining that image, I can offer the following example, which may qualify as being of historical interest.<br><br>
The first hint that there was another $K(\pi,1)$ for finite type Artin groups of rank bigger than two (apart from the well-known quotient of the permutahedron) was in [my paper <a href="./papers/1994 - Brady - Automatic structures on Aut(F_2).pdf">Automatic structures on $\text{Aut}(F_2)$</a>], where the 2-complex has fundamental group which is commensurable with the four strand braid group mod its centre.   The sixteen 2-cells correspond to the sixteen factorisations of a 4-cycle as a product of three transpositions.
<br>
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Thomas Brady
</div>
<center><a href="./papers/1994 - Brady - Automatic structures on Aut(F_2).pdf"><img data-src="./img/Brady F2.png" height="300"></a>$~~~~~$<a href="./papers/2002 - Brady, Watt - K(pi,1)'s for Artin groups of finite type.pdf"><img data-src="./img/Brady - Title.png" height="300"></a></center>

<u>Another word from Jean **MICHEL**</u>
> <div style="font-size: .5em; width:text-width; height:250px;overflow:auto">
In  1999, Joan  Birman gave  a talk  in Luminy  on the  recently discovered Birman-Ko-Lee  monoid (now called the dual monoid  in type A). This was the first  time I  met her.  At that  time I  was interested in the Artin braid
groups  since I had conjectures  with Broué and Malle  on the cohomology of
Deligne-Lusztig  varieties  stating  that  the  commuting  algebra  of  the
reductive  group acting  on that  cohomology should  be a  cyclotomic Hecke
algebra.  In particular we needed to  show that the centralizer of periodic
braids are braid groups for the centralizer of the image of these braids in
the  Weyl group, which  are Springer regular  elements. A very special case
for  the Deligne-Lusztig variety attached to  a coxeter element is that the
centralizer  of the  lift $\delta$  of a  Coxeter element  to the braid group
would  be  the  cyclic  group  generated  by  the  Coxeter element. We were
struggling  trying to  prove that.  I was  the only  person in our group to
attend the braid conference in Luminy.<br><br>
During  Joan's talk I realized that  the Birman-Ko-Lee monoid was a Garside
monoid  (the terminology was "small Gaussian  monoid" at the time, from the
just published paper of Dehornoy and Paris) and that \delta was its Garside
element.  And  it  is  easy  to  compute  the  fixed  points of the Garside
automorphism induced by a Garside element!<br><br>
Back  in Paris I explained  this to David Bessis  and François Digne and it
gave rise to the paper <a href="">Springer theory in braid groups and the
Birman-Ko-Lee monoid</a>, where  we define a dual monoid also in  type B. Soon after, David found the correct   definition  for  the  dual  monoid  in  other  Artin  types.  The conjectures  on the centralizers of periodic  braids were proved in general only  later. They are consequences  of the beautiful paper  of David on the $K(\pi,1)$ property.
>$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Jean Michel
</div>

<center><a href="./papers/2002 - Bessis, Digne, Michel - Springer theory in braid groups and the Birman-Ko-Lee monoid.pdf"><img data-src="./img/Springer - Title.png" height="300"></a>$~~~~~$<a href="./papers/2003 - Bessis - The dual braid monoid.pdf"><img data-src="./img/Bessis - Title.png" height="300"></a>

<u>A word from Hugh **THOMAS**</u>
> <div style="font-size: .5em; width:text-width; height:150px;overflow:auto">
If I remember correctly, I asked Colin [Ingalls] to tell me about quivers because I had overheard Frédéric Chapoton say the word at the 2005 AIM meeting (in casual conversation). (And Colin had already mentioned quivers when we had had a chat after I had come to New Brunswick.) That's not much of a story, but that is how that paper of ours got started.<br><br>
Once you look at the sinple reflections of a noncrossing partition as indecomposables in $A_3$, it's pretty evident that the "crossing" is failure of those objects to form the simple objects of an abelian subcategory, and then that just keeps working.$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Hugh Thomas
</div>

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<u>A word from Christian **STUMP**</u>
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### <u> Selected References</u> 
<center>
<a href="./papers/2000 - Simion - Noncrossing partitions.pdf"><img data-src="./img/Simion - Title.png" height="300"></a><a href="./papers/2006 - McCammond - Noncrossing partitions in surprising locations .pdf"><img data-src="./img/McCammond - Title.png" height="300"></a>
<a href="./papers/2006 - Reading - Cambrian lattices.pdf">
      <img data-src="./img/Reading 2.png" style="height:300px;"></a>
      <a href="./papers/2007 - Reading - Sortable elements and Cambrian lattices.pdf"><img data-src="./img/Reading 1.png" height="300"></a>
      <a href="./papers/2010 - Reading, Speyer - Sortable elements in infinite Coxeter groups.pdf"><img class="d-block w-100" src="./img/Reading 3.png" style="height:300px;"></a><a href="./papers/2009 - Ingalls, Thomas - Noncrossing partitions and representations of quivers.pdf"><img data-src="./img/Ingalls Thomas Title.png" height="300"></a>

</center>

<u>A word from Brendon **RHOADES**</u>
> <div style="font-size: .5em; width:text-width; height:500px;overflow:auto">
My introduction to Catalan theory came through parking theory. With origins coming from labelling regions of the Shi arrangement (and its relatives), Drew Armstrong, Vic Reiner, and I introduced "noncrossing parking functions" associated to any (real, irreducible) reflection group $W$. These are equivalence classes $[w, X]$ of pairs $(w, X)$, where ... <br>
> - $w$ is an element of the group $W$ <br>
> - $X$ is a $W$-noncrossing flat, and<br>
> - $(w, X) ~ (w' X)$ whenever we have the equality of cosets $w W_X = w' W_X$.<br><br>
The set $\mathrm{Park}^{\mathrm{NC}}_W$ of $W$-noncrossing parking functions admits an action of the product group $W \times C$ via<br>
$$(v, c^d) . [w, X] := [v w c^{-d}, c^d X].$$
Drew, Vic, and I formulated the "weak" parking conjecture, which gives a character formula for this action related to the spectral data of $w$ acting on the reflection representation $V$.<br>
Our attack on the weak parking conjecture was algebraic. Suppose $W$ has rank $r$ and let $\mathbb{C}[V]$ be the coordinate ring of the reflection representation of $W$. Remarkable work of Kraft, Etingof, Griffeth, Gordon ... tied to Cherednik algebras guarantees the existence of certain 'magical' polynomials $\theta_1, \ldots , \theta_r$ such that
$\theta_1, \dots, \theta_r$ are homogeneous of degree h+1 and form a regular sequence (a "homogeneous system of parameters", or h.s.o.p.), and
the linear span of $\theta_1, \ldots , \theta_r$ is $W$-stable and carries the reflection representation $V^*$ of W.
<br><br>
In this setting, the quotient $\mathbb{C}[V]/(\theta_1, \ldots , \theta_r)$ is naturally a $W \times C$-module, and is uniformly understood to have the same character as the formula in the weak parking conjecture. 
<br><br>
Our goal was to prove that the combinatorial representation $\mathrm{Park}^{\mathrm{NC}}_W$ and the algebraic quotient $\mathbb{C}[V]/(\theta_1, \ldots , \theta_r)$ were isomorphic as $W \times C-modules$. One issue is that the quotient $\mathbb{C}[V]/(\theta_1, \ldots , \theta_r)$ corresponds to a fat point of multiplicity $(h+1)^r$ supported at the origin, and is therefore not so combinatorial in nature. In order to combinatorailize the situation, we used deformation theory: if $x_1, \ldots , x_r$ is a basis of $V^*$ for which the map sending $x_i$ to $\theta_i$ is $W$-equivariant, one can consider the "parking locus"
$$Z_W := \text{zero set of } ( \theta_1 - x_1, \ldots , \theta_r - x_r) \text{ in } V.$$
Etingof proved that $Z_W$ is indeed a reduced locus for well-chosen h.s.o.p.'s. The orbit harmonics technique of deformation theory implies that $Z_W$ carries the same $W \times C$-structure as $\mathrm{C}[V]/(\theta_1, \ldots , \theta_r)$. The "strong" parking conjecture asserts that there is a $W \times C$-equivariant bijection between $\mathrm{Park}^{\mathrm{NC}}_W$ and $Z_W$.
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The orbit harmonics technique was introduced by Kostant and has seen influential attention by Garsia-Procesi, Garsia-Haiman, N. Bergeron-Gagnon, and others. It has played a hugely important role in my research, leading to to quotient rings for delta operators, quantum Donaldson-Thomas invariants, the Viennot shadow avatar of RSK, and more. I am forever grateful to Catalan theory for introducing me to this remarkable technique.
	>$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Brendon Rhoades
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