My aim is to explain some ideas relating
combinatorics and partial orders
isolated singularities of polynomial functions
derived categories
⚠️ Some pieces are missing or outside my reach.
Derived categories serve as a meeting point between combinatorics and singularity theory.
In the field of enumerative combinatorics, one aims at counting things.
Sometimes, one gets closed formulas, of various sorts.
Maybe the nicest sort is a product formula.
For example, all lattice paths from $(0,0)$ to $(m,n)$ using only steps $(1,0)$ and $(0,1)$.
These are counted by the binomial coefficient $\binom{m+n}{m}$.
This formula can be written as follows
$$\binom{m+n}{m} = \prod_{i=1}^{m} \frac{m+n+1-i}{i},$$which has the general shape
$$\prod_{e \in E} \frac{D - e}{e},$$for some multi-set $E$ and integer $D$ (here $E = \{1,2,\ldots,m\}$ and $D = m+n+1$).
Fact: Many famous enumeration results in combinatorics involve formulas of this precise shape !
und so weiter...
No obvious general reason is known for these formulas to exist. Some partial reasons explain some cases.
For example, the number of clusters in a cluster algebra is given by
$$ \prod_{i=1}^{m} \frac{h+d_i}{d_i} ,$$where $h$ and the $(d_1,\dots,d_m)$ are the Coxeter number and the degrees of the associated Weyl group.
Using the symmetry of the degrees $d_i \leftrightarrow h+2 - d_{\varphi(i)}$ for some bijection $\varphi$, this is the same as
$$ \prod_{i=1}^{m} \frac{2 h + 2 - d_i}{d_i} ,$$which has exactly the expected shape with $D = 2h+2$.
We will look at isolated quasi-homogeneous singularities of functions $f : \mathbb{C}^m \to \mathbb{C}$.
These were studied by Milnor, in famous and classical works.
Recall that $z = (z_1,\ldots,z_m) \in \mathbb{C}^m$ is a singular point of $f$ if all partial derivatives of $f$ vanish at $z$.
The word "isolated" means that $f$ has an isolated singular point, that we assume to be $0 \in \mathbb{C}^m$.
We will also assume that $f(0) = 0$.
The word "quasi-homogeneous" means that there exists integers $d_1,\ldots,d_m$ and $D$ such that:
the total degree of $f$, when giving weight $d_i$ to the variable $z_i$, is given by $D$.
Example: $f = x^3 + y^4$ has total degree $12$ if $x$ has weight $4$ and $y$ has weight $3$.
So every quasi-homogeneous isolated singularity comes with the data of $(d_1,\ldots,d_m),D$.
Conversely, fix $(d_1,\ldots,d_m),D$ and pick $f$ as a generic quasi-homogeneous polynomial w.r.t. this data.
Fact: There exists necessary and sufficient conditions on $(d_1,\ldots,d_m),D$ in order to ensure that $f$ has an isolated singularity.
I will not need these rather technical conditions. We always assume that they hold.
Fact (A): The product $\prod_{i=1}^{m} \frac{D-d_i}{d_i}$ is an integer.
Fact (B): The product $\prod_{i=1}^{m} \frac{[D-d_i]_q}{[d_i]_q}$ is a polynomial in $\mathbb{N}[q]$.
Here $[d]_q$ denotes the $q$-integer $1+q+q^2+\cdots+q^{d-1}$.
Both facts are not clear a priori. They only belong clearly to $\mathbb{Q}$ and $\mathbb{Q}(q)$.
A quick little piece of geometry now, from Milnor
Choose $f$ with quasi-homogeneous isolated singularity. Let $B$ be a small-enough ball around $0 \in \mathbb{C}^m$.
the fibers $f^{-1}(z) \cap B$ for $z$ in a small circle around $0 \in \mathbb{C}$ are all diffeomorphic
they form a locally-trivial fibration over the small circle
they have the homotopy type of a bouquet of $\mu$ spheres of dimension $m-1$
Sketch picture on the board if there is a board.
Fact (A2): the integer $\mu = \prod_{i=1}^{m} \frac{D-d_i}{d_i}$ (Milnor number of the singularity)
Fact (B2): the polynomial $\prod_{i=1}^{m} \frac{[D-d_i]_q}{[d_i]_q}$ describes a filtration on the homology of the fibers.
Because fibers are bouquets 💐 of spheres 🌎 of the same dimension, only one interesting homology group $H^{m-1}$ isomorphic to $\mathbb{Z}^{\mu}$.
Turning once around the small circle and following cycles by local triviality, one obtains a linear map
$H^{m-1} \to H^{m-1}$ which is called the monodromy map of the singularity.
You can think of the monodromy as a $\mu \times \mu$ matrix of integers.
Main idea : relate product formulas in combinatorics to Milnor formula for Milnor number $\mu$ of singularities
HOW ? Using derived categories
on the combinatorial side, enrich combinatorial objets with partial orders
and consider modules over their incidence algebras over a field
on the singularity side, consider a categorification $\mathscr{D}_{\operatorname{Mil}}$ of the Milnor fiber homology and monodromy
This $\mathscr{D}_{\operatorname{Mil}}$ should be a triangulated category, recovering the geometric data when passing to $K_0$.
⚠️ WARNING: Apparently, this kind of category has not yet been constructed in any kind of generality. I am not sure.
This $\mathscr{D}_{\operatorname{Mil}}$ should be Something Like a Directed Fukaya Category 🦄, whatever it is
Here some hand-waving about A-Model, B-model, mirror symmetry ?
Suppose that you have a familly of combinatorial objects $(P_n)_n$ counted for each index $n$ by a combinatorial formula of the shape
$$\prod_{e \in E} \frac{D - e}{e}$$for some multi-sets $E$ and integers $D$ depending on the index $n$ in some regular way.
(Example : Dyck paths and Catalan numbers)
THEN 🎁
There should exist partial orders $\leq$ on the combinatorial objects such that the derived category $\mathscr{D}_P$ of modules over the incidence algebra of $(P_n, \leq)$ is triangle-equivalent to the derived category $\mathscr{D}_{\operatorname{Mil}}$ attached to the quasi-homogeneous singularity of a generic quasi-homogeneous polynomial with weights $E$ and total weight $D$.
(Example: the natural partial order on Dyck paths by inclusion)
Claim: all the derived categories involved should be fractional Calabi-Yau.
Let $T$ be triangulated category with finite-dimensional $\operatorname{Hom}$ spaces over a field
A Serre functor on $T$ is an auto-equivalence of $T$ such that
$$\operatorname{Hom}(X,Y)^{*} \simeq \operatorname{Hom}(Y,S X)$$(functorially in both arguments)
This name comes from the Serre duality functor on coherent sheaves in algebraic geometry. Unique up to isomorphism.
The category $T$ is Calabi-Yau if $S$ is isomorphic to a shift functor $[D]$.
This names comes from the properties of coherent sheaves on Calabi-Yau manifolds.
The category $T$ is fractional Calabi-Yau if a power of $S$ is isomorphic to a shift functor. (Kontsevich, around 2000)
meaning that $S^q \simeq [p]$ for some integers $p$ and $q$. Abusingly, $p/q$ is called the Calabi-Yau dimension.
some examples come from algebraic geometry : pieces in semi-orthogonal decompositions of derived categories coming from Fano manifolds
derived categories of representations of Dynkin quivers (types $\mathbb{ADE}$)
some examples from singularity theory using categories of matrix factorisations
Combinatorics of posets-with-product-formula as a new source of examples !
On the geometry side, the expected categorification $\mathscr{D}_{\operatorname{Mil}}$ of Milnor's fibration should be such that
On the combinatorial side, the derived category $\mathscr{D}_P$ of modules over an incidence algebra of a partial order $P$ has an Auslander-Reiten functor $\tau$ (equivalent to having a Serre functor $S$)
Suppose you have combinatorial objects counted by a product formula.
Suppose moreover that you have found partial orders on these objects.
How to convince yourself that they are "good" in the sense of our motto 🎁 ?
💡 IDEA : compare the characteristic polynomials of monodromy !
Product formula $\implies$ weights $E$ and degree $D$ $\implies$ formula for characteristic polynomial (Milnor-Orlik)
So one has a guess for the characteristic polynomial, to compare with the one from the partial order (called the Coxeter polynomial of $P$)
If they match, one can hope to be on a good track !
Consider the partial orders on Dyck paths by inclusion (being always below)
In size $3$, there are $5$ Dyck paths.
The general formula is the Catalan number $\frac{1}{n+1}\binom{2n}{n}$.
For $n=3$, this gives the product formula
$$5 = \frac{6}{2}\frac{5}{3}$$so that $E = \{2,3\}$ and $D = 8$.
From this, one finds that the monodromy of a generic singularity has char. polynomial $t^5 - t^4 + t - 1$.
On the other hand, one computes the matrix of the Auslander-Reiten translation $\tau$ and finds
$$ \left(\begin{array}{rrrrr} 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 & -1 \\ 0 & 1 & 0 & 0 & -1 \\ -1 & 1 & 1 & 0 & -1 \\ 0 & 0 & 0 & 1 & -1 \end{array}\right) $$which has the same char. polynomial (up to technical details about shifts and signs).
It can very well happen that two famillies of combinatorial objects share the same product formula.
It can also happen that one finds different partial orders both being good w.r.t this product formula.
In this case, one should expect the derived categories of posets to be triangle-equivalent and fractional Calabi-Yau. This can be proved by purely algebraic means.
Prototypical examples : Dyck words for inclusion and binary trees for rotation (Tamari order)
both counted by Catalan numbers, both apparently sharing the same char. polynomials.
It is known (Rognerud) that Tamari orders are indeed fractional Calabi-Yau.
Not known yet for Dyck paths under inclusion.
Sometimes not easy to find good partial orders for which the motto 🎁 would work
good candidate for the famous formula for alternating sign matrices, partial order found by J. Striker.
no candidate known for the formula by Tutte counting intervals in the Tamari lattices
no candidate known for the formula enumerating 2-stack sortable permutations
So there remains nice things to discover in the wild out there.
Thom-Sebastiani sum of singularities : $f$ and $g$ with disjoint variables, consider $f+g$
in terms of weight data $(d_1,\ldots,d_m);D$ and $(e_1,\ldots,e_n);E$, this means:
For example $(1);3$ and $(1);4$ together give $(4,3);12$
The the monodromy of Milnor fibers is the tensor product of monodromies
Then the associated Fukaya-style categories shoud be tensor product of the smaller Fukaya-style categories.
On the combinatorial side, for the cartesian product of posets, one also get that the derived category is the tensor product of categories for factor posets.
$\implies$ if you can factorise the weight data, you should be able to factorise the poset (up to derived equivalence)