Beyond the g-conjecture – algebraic combinatorics of cellular spaces I

Combinatorics and more 2018-08-02

The g-conjecture for spheres is surely the one single conjecture I worked on more than on any other, and also here on the blog we had a sequence of posts about it by Eran Nevo (I,II,III,IV). Here is a great survey article on the g-conjecture by Ed Swartz 35 Years and Counting and slides of a great lecture by Lou Billera “Even more intriguing, if rather less plausible…”.

June Huh talked about the conjecture on Numberphile attracting 183,893 views and counting. With hundreds of thousands new people interested in the g-conjecture and some highly capable teams of researchers who were interested even before the video, it is safer to say that the solution of the conjecture is imminent and this is going to be exciting! So I want to say a few things about the “G-program”, questions that go beyond the g-conjecture. (OK, I realize I need to split it into parts and also that each of the items deserves a separate post in the future.)

I wrote a post to suplement Eran Nevo’s posts on how the g-conjecture came about, and here is a very brief summary:

Simplicial polytopes: Leading to the g-theorem we can start with Euler’s theorem, and its high dimensional extensions for general polytopes. Next follow the Dehn-Sommerville relations for simplicial polytopes. Here are some main landmarks toward the g-conjecture and its solution for simplicial polytopes.

The upper bound theorem (UBT, conjectured by Motzkin, 1957, proved by Peter McMullen 1970),
The lower bound theorem (LBT, conjectured by Grunbaum in the 60s, proved by Barnette 1970),
In 1970, the generalized lower bound conjecture (GLBC) was conjectured by McMullen and Walkup, and the g-conjecture was proposed by McMullen.
In 1979, the sufficiency part of the g-conjecture was proved by Billera and Lee.
Also in 1979, the necessity part of the g-conjecture was proved by Stanley based on the hard Lefschetz theorem for toric varieties.
In 1993 Peter McMullen found a direct geometric proof for the case of simplicial polytopes.
The GLBC (now, GLBT) consists of the linear inequalities of the g-conjecture and an additional part about cases of equality. The equality part was proved in 2012 by Murai and Nevo (see this post).

Triangulated spheres: All of the above are conjectured to hold for simplicial spheres (and even to homological spheres in the most inclusive sense). Barnette’s proof for the LBT applies to triangulated spheres. The UBT was proved for spheres by Stanley (1975) pioneering the connection with commutative algebra. The g-conjecture for triangulated spheres is still open, and this is what we refer to these days as the “g-conjecture”.

Algebraic tools: For the connection with commutative algebra and algebraic geometry pioneered by Stanley, see this post. The crucial linear algebraic property behind the g-conjecture is a linear algebraic statement called the Lefschetz property that is a consequence of the Hard Lefschetz Theorem (when it applies). McMullen’s 1993 proof validates the Lefschetz property in greater generality (for arbitrary simplicial polytopes rather than rational simplicial polytopes) via an inductive argument that relies on “Pachner moves”. Certain Hodge-Riemann inequalities also played an important role in McMullen’s proof. Hodge-Riemann’s inequalities play a crucial role also in the recent solution of Rota’s conjecture by Adiprasito, Huh and Katz. (See this post and this beautiful Quanta Magazine article by Kevin Hartnett.)

In this extra footage for his Numberphile video (viewes by 10,000) June mentions the hard Lefschetz theorem.

Eran Nevo (left) and Karim Adiprasito

Going Beyond the g-conjecture: The “G-Program”

Let $K$ be a class of cellular $(d-1)$-dimensional spheres.

We want to define $d$ -parameters $h_0(K),h_1(K)\dots h_{d}(K)$ forming the $h$ -vector of $K$ , that satisfies

(1) latex h_i(K)=h_{d-i}(K)$,

(2) $1=h_0(K) \le h_1 (K)\le \dots h_[d/2] (K)$ . In other words if we let $g_i=h_1-h_{i-1}$ then $g_i \ge 0$ . (The $g_i$ are the g-numbers giving the conjecture its name.)

(3) Some interesting non-linear inequalities in the spirit of those for the original conjecture.

If this program is not general enough we would like to consider arbitrary manifolds without boundary or even pseudomanifolds without boundary, to make some adaptation to allow boundary, and even to allow the “cells” to be rather exotic. We also want to understand the equality cases for the inequalities and how the h-vectors reflect the combinatorics, topology, and algebra of the cellular space K.

Some classes of cellular spheres: The class of simplicial spheres (and polytopes) contains the classes of flag spheres (and polytopes) and completely balanced spheres (and polytopes). The class of regular CW spheres with the intersection property, (including general polytopes) contains the classes of cubical spheres (and polytopes) and centrally symmetric spheres (and polytopes). Regular CW spheres without the intersection property include those coming from Bruhat intervals of Coxeter groups. (The class of zonotopes is also very important but its connections to the present story are more subtle.)

From the algebraic geometry side, simplices correspond to complex projectice spaces, the h-vectors for simplicial polytopes correspond to Betti numbers that measure how more complicated a general smooth (toric) variety is. For general (rational) polytopes, you get more complexity in terms of complicated singularities, and for objects like intervals in Bruhat orders you have even more complicated singularities. The varieties exist only for small fragments of the pictures and beyond that you need to do some sort of “algebraic geometry without varieties”.

A) The original g-conjecture: simplicial polytopes and simplicial spheres

Again, let me refer the reader to the four posts by Eran Nevo (I,II,III,IV). I will repeat the definition of h-vectors and g-vectors at the end of the post.

Problem 1: Prove the g-conjecture for triangulated sphere.

Again, the most promising avenue towards a proof is by proving the Lefschetz property for face rings associated with triangulated spheres.

Problem 2: Understand the cases of equality for the Macualay (non-linear) inequalities?

Problem 3: Show that the gap in the inequalities tends to infinity for every sequence of simplicial polytopes tending to a smooth body. Formulate and prove an extension to spheres.

In the polytope case this problem was settled. This has been proved (for polytopes, see this paper and this post) by Karim Adiprasito, Eran Nevo, and José Alejandro Samper for the linear inequalities (see picture below). The result, referred to as the “geometric LBT” asserts that for a sequence of simplicial polytopes $P_n$ tending to a smooth body, $g_i(P_n) \to \infty.$ More recently, Karim Adiprasito managed to prove it for the nonlinear inequalities. For triangulated spheres, finding the relevant notion of limits for triangulations would be a place to start.

The Lefschetz property (when it holds) allows to associate to every simplicial sphere S, a (shifted) order of monomials M(S) with $g_i$ monomials of degree $i$ , $i=1,2,\dots,[d/2]$ of degree $[d/2]$ . There is also a construction (Kalai, 1988) that associates a triangulated sphere S(M) (called “squeezed sphere”) with every such M. Satoshi Murai proved that M(S(M))=M.

Problem 4: Study M(S)

B) General polytopes: The toric h-vector and g-vector

Intersection homology allows the definition of h- and g- polynomials for general polytopes. Those are linear combinations of flag numbers. The post Euler’s formula, Fibonacci, the Bayer-Billera theorem, and Fine’s cd-index is very relevant.

Again, I will repeat a definition of h-vectors and g-vectors at the end of the post. HLT for intersection homology shows the nonnegativity of the g-polynomials for rational polytopes. Kalle Karu (relying on works by Barthel, Brasselet, Fieseler and Kaup and by Bressler and Luntsand and on McMullen’s 1998 proof) proved it for general polytopes!

Problem 5: Show that the toric g-numbers are nonnegative for every strongly regular CW-sphere. (In particular for all polyhedral spheres.)

Recall that a regular CW complex is a CW-complex where the closure of an open cell is homeomorphic to a close ball. Regular CW spheres such that the intersection of two faces is a face are also called strongly regular CW complexes.

A big open problem which is open even for polytopes is

Problem 6: Show that the toric g-numbers form an M-vector.

Problem 6 is of great interest also for rational polytopes. The difficulty is that intersection homology does not admit a ring structure. One approach is to introduce some additional ring structure while another approach is to try to derive the M-inequalities from weaker algebraic or combinatorial conditions.

Problem 7: Understand the cases of equality for the linear inequalities and the non linear inequalities.

Recent advances by Adiprasito and Nevo in their paper QGLBT for polytopes toward the equality case for the toric GLBC. They also extended the geometric LBT to general polytopes.

Problem 8 (Jonathan Fine): Extend the toric g-numbers to a Fibonacci number of sharp nonnegative parameters, based, perhaps, on a Fibonnaci number of homology groups.

Interesting and rather mysterious duality relations were discovered for g-numbers. They were found to be related to Koszul duality and to Mirror symmetry. See this recent post. $\gamma (P)$ of that post is $g_2(P)$ .

Problem 9 : Understand further the connections between toric g-numbers and related invariants of polytopes and their dual.

Closely related issues are: flag vectors and the Bilerra-Bayer theorem (see this post), nonnegative flag inequalities, the cd-index, FLAGTOOL, and Braden-MacPherson theorem. An excellent 2006 paper on toric h- and g- numbers and related combinatorics and algebraic-geometry is by Tom Braden: Remarks on the combinatorial intersection cohomology of fans.

Problem 10: Understand (all) linear inequalities among flag numbers of $d$ -polytopes and polyhedral $d$ -spheres.

Warning: I used Whitehead notion “strongly regular CW complexes”, and “regular CW complexes with the lattice property” and “regular CW-complexes with the intersection property” for the same mathematical object. Sorry.

C) Regular CW-spheres and Kazhdan-Lusztig polynomials

What happens when you give up also the lattice property? For Bruhat intervals of affine Coxeter groups the Kazhdan Luztig polynomial can be seen as subtle extension of the toric g-vectors adding additional layers of complexity. Of coursem historically Kazhdan-Lustig polynomials came before the toric g-vectors. (This time I will not repeat the definition and refer the readers to the original paper by Kazhdan and Lustig, this paper by Dyer and this paper by Brenti. Caselli, and Marietti.) It is known that for Bruhat intervals with the lattice property the KL-polynomial coincide with the toric g-vector. Can one define h-vectors for more general regular CW spheres?

Problem (fantasy) 12: Extend the Kazhdan-Luztig polynomials (and show positivity of the coefficients) to all or to a large class of regular CW spheres.

This is a good fantasy with a caveat: It is not even known that KL-polynomials depend just on the regular CW sphere described by the Bruhat interval. This is a well known conjecture on its own.

Problem 11: Prove that Kazhdan-Lustig polynomials are invariants of the regular CW-sphere described by the Bruhat interval.

A more famous conjecture was to prove that the coefficients of KL-polynomials are non negative for all Bruhat intervals and not only in cases where one can apply intersection homology of Schubert varieties associated with Weil groups. (This is analogous to moving from rational polytopes to general polytopes.) In a major 2012 breakthrough, this has been proved by Ben Elias and Geordie Williamson following a program initiated by Wolfgang Soergel.

There is vast literature on KL-polynomials further extensions and combinatorial aspects. The combinatorics of regular, but not strongly regular CW complexes is again related to the cd-index and let me also mention that there is interesting combinatorics (and a good opportunity for the G-program and fantasy 12) for simplicial posets, namely when you insist on lower intervals to be Boolean but give up the lattice property.

D) Cubical polytopes and spheres: Adin’s h-vectors

A cubical polytope (sphere) is a polytope all whose faces are combinatorial cubes. Cubical complexes are important and mysterious objects. (They play a crucial role in some recent developments in 3D topology, see here.) Ron Adin defined h-numbers and formulated a “GLBC ineqequalities for cubical polytopes (and spheres). Again, I will repeat Adin’s definition of h-vectors and g-vectors at the end of the post.

Problem 13: Prove Adin’s conjecture for cubical polytopes and spheres.

A recent paper by Adin, Kalmanovich, and Nevo On the cone of f -vectors of cubical polytopes shows that if Adin’s conjecture is valid, it describes the full cone spanned by linear inequalities among face number of cubical d-polytopes.

Problem 14: Explore an UBT for cubical polytopes and non-linear inequality for Adin’s numbers.

Problem 15: Extend (in the toric spirit) Adin’s invariant to polytopes and spheres with the property that every two-dimensional face has at least four edges (or just an even number of edges).

Coming next on the G-program:

E) Centrally-symmetric polytopes and spheres;
F) Flag polytopes and spheres – the Charney-Davis conjecture and the $\gamma$ -conjecture;
G) Completely balanced polytopes and spheres;
H) Polytope pairs and polytopes with one non-simplex facet;
I) The Batchi-Datta conjecture;
J) Sharper versions of the generalized lower bound inequalities and further applications of the Lefschetz property.
K) Stanley’s local theory
L) Simplicial posets and ASL.
M) Minkowski sums of polytopes;
N) Section of a given polytope;
O) Integer points in polytopes and associated polynomials
P) Grunbaum’s conjecture and the GUBT (a related old post); (Let me mention that Karim Adiprasito reported recently on progress on the Grunbaum’s conjecture which is related to the g-conjecture.)
Q) The Welzl-Wagner framework and early continuous analogs;
R) triangulations of manifolds and pseudomanifolds.

I am sure that I missed, overlooked, forgot, or wasn’t aware of several things that I should mention. Please comment here or alert me about them. Also most of the problems I described are on the basic combinatorics level of the theory and Karim promised to contribute some problems on the algebraic level for a later post in this series.

STANLEYLAND-enumerative algebraic combinatorics

Some definitions

General polytopes: Toric h and g

Consider general d-polytopes. For a set $S \subset$ {0,1,2,…,d-1}, $S=$ { $i_1,i_2,\dots,i_k$ } , $i_1<i_2< \dots$ , define the flag number $f_S$ as the number of chains of faces $F_1 \subset F_2 \subset \dots F_k$ , where $\dim F_j=i_j$ . If $P$ is a $d$ dimensional polytope and $Q$ is an $e$ dimensional polytope, their direct sum $P \oplus Q$ is obtained by embedding P and Q in two orthogonal subspaces such that the origin is an interior point of both P and Q, and taking the convex hull. The free join $P*Q$ is the convex hull of P and Q when they are embedded in two skew affine spaces of dimensions d and e in a (d+e+1)-dimenional space.

For the original recursive definition of toric h-vectors see Stanley’s 1987 paper Generalized h-vectors, intersection cohomology of toric varieties, and related results. A simple axiomatic definition of the toric $h$ -vectors (taken from my 1988 paper A new basis of polytopes) is the following: We consider two polynomials $h[P](x)$ and $g[P](x)$, with the properties

(A1) $h[P](x)=\sum_{0\leq i\leq d}h_i(K)x^{d-i},$ and $g[P](x)=\sum_{0\leq i\leq [d/2]}g_i(K)x^{d-i}$ , where $g_i=h_i-h_{i-1}$ .

(A2) $h[P](x)=g[P](x) =1$ if $P$ is a point.

(A3) The coefficients $h_i$ of $h[P](x)$ are linear combinations of flag numbers.

(A4) $h[P \oplus Q](x)=h[P](x)h[Q](x).$

(A5) $g[P * Q](x)=g[P](x)g[Q](x).$

(A1)-(A5) determine uniquely the polynomials h and g. In the simplicial case all flag numbers are determined by ordinary face numbers and this gives the definition described below.

Cubical polytopes: Adin’s short h-vector and (long) h-vector

Adin’s short h-vector is defined as follows:

$h^{sc}[Q](t)=\sum_{i=0}^{d-1} h_i^{sc}(Q)t^j=\sum_{j=0}^{d-1}f_j(Q)(2t)^j(1-t)^{d-1-j}.$

Adin’s (long) h-vector is defined by $h_0^c=2^d$ , and $h_i^{sc}=h_i^c+h_{i+1}^c$ , for $0 \le i \le d-1$ .

The short and long $g$ -vectors are defined by taking differences as in the simplicial case. $g_0^{sc}=h_o^{sc}=f_0$ , $g_i^{sc}=$ $h_i^{sc}-$ $h_{i-1}^{sc}$ for $1\le i\le [(d-1)/2]$ ; $g_0^c=h_o^c=2^{d-1}$ . $g_i^{c}=$ $h_i^{c}-$ $h_{i-1}^{c}$ for $1\le i\le [d/2]$ .

The simplicial case: h-vectors g-vectors, and the g-conjecture

The $f$ –vector (face vector) of a cellular complex $K$ is $f(K)=(f_{-1},f_0,f_1,...)$ where $f_{i}$ is the number of $i$ -dimensional faces of $K$ . For example, if $K$ is the $(d-1)$ -simplex then $f_{i-1}(K)=\binom{d}{i}$ . Let $K$ be a $(d-1)$ -dimensional simplicial complex. the $h$ –vector of $K$ is defined by

$\sum_{0\leq i\leq d}h_i(K)x^{d-i}= \sum_{0\leq i\leq d}f_{i-1}(K)(x-1)^{d-i}.$

Let $g_0(K):=h_0(K)=1$ , $g_i(K):=h_i(K)-h_{i-1}(K)$ for $1\leq i\leq \lfloor d/2\rfloor$ . $g(K):=(g_0(K),...,g_{\lfloor d/2\rfloor}(K))$ is called the $g$ -vector of $K$ . The Dehn-Sommerville relations state that when $K$ is a sphere $h_i(K)=h_{d-i}(K)$ for every $0\leq i\leq d$ . (This result can be proved combinatorially, for the larger family of Eulerian posets, and, for rational simplicial polytopes it reflects Poincare duality for the associated toric variety.)

We say that a vector $h$ is an $M$ -vector ( $M$ for Macaulay) if it is the $f$ -vector of a multicomples, i.e. of a collection of multisets (elements can repeat!) closed under inclusion. For example, $h=(1,1,1)$ is an $M$ -vector, as is demonstrated by the multicomplex $\{1=x^0,x,x^2\}$ , written in monomial notation – the exponent tells how many copies of $x$ to take. Macaulay gave a numerical characterization of such vectors. (The proof uses compression – see this post for a general description of the method.)

The $g$ -conjecture:

The following are equivalent:

(i) The vector $(f_0,f_1,\dots , f_{d-1})$ is the $f$ -vector of a simplicial $d$ -polytope

(ii) The vector $(f_0,f_1,\dots , f_{d-1})$ is the $f$ -vector of a triangulated $(d-1)$ -sphere.

(iii)

(a) $h_i=h_{d-i}$ for every $i$

(b) $g(K)$ is an $M$ -vector.

Last observation: explained to the general public, the most confusing aspect of the story is when you draw a triangle and refer to it as a “sphere”.

A description of the Adiprasito, Nevo, and Samper paper.

Our chair Elon Lindenstrauss promised to ease the financial burdon caused by my habit of paying for Karim Adiprasito coffee’s once after every breakthrough, so I started to collect the receipts.