Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial - PowerPoint PPT Presentation

Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries June Huh Institute for Advanced Study and Princeton University with Karim Adiprasito and Eric Katz June Huh 1 / 26

A graph is a ✶ -dimensional space, with vertices and edges. Graphs are the simplest combinatorial structures. June Huh 2 / 26

✎ ✎ ✎ ✎ ✤ ● ✭ q ✮ ❂ ✶ q ✹ � ✺ q ✸ ✰ ✽ q ✷ � ✹ q ❀ ✤ ● ✭✷✮ ❂ ✵ ❀ ✤ ● ✭✸✮ ❂ ✻ ❀ ✿ ✿ ✿ ✤ ● ✭ q ✮ ● ❛ ✷ ✐ ✕ ❛ ✐ � ✶ ❛ ✐ ✰✶ ✐ Hassler Whitney (1932): The chromatic polynomial of a graph ● is the function ✤ ● ✭ q ✮ ❂ ✭ the number of proper colorings of vertices of ● with q colors ✮ ✿ June Huh 3 / 26

✤ ● ✭ q ✮ ● ❛ ✷ ✐ ✕ ❛ ✐ � ✶ ❛ ✐ ✰✶ ✐ Hassler Whitney (1932): The chromatic polynomial of a graph ● is the function ✤ ● ✭ q ✮ ❂ ✭ the number of proper colorings of vertices of ● with q colors ✮ ✿ Example ✎ ✎ ✎ ✎ ✤ ● ✭ q ✮ ❂ ✶ q ✹ � ✺ q ✸ ✰ ✽ q ✷ � ✹ q ❀ ✤ ● ✭✷✮ ❂ ✵ ❀ ✤ ● ✭✸✮ ❂ ✻ ❀ ✿ ✿ ✿ What can be said about the chromatic polynomial in general? June Huh 3 / 26

Hassler Whitney (1932): The chromatic polynomial of a graph ● is the function ✤ ● ✭ q ✮ ❂ ✭ the number of proper colorings of vertices of ● with q colors ✮ ✿ Example ✎ ✎ ✎ ✎ ✤ ● ✭ q ✮ ❂ ✶ q ✹ � ✺ q ✸ ✰ ✽ q ✷ � ✹ q ❀ ✤ ● ✭✷✮ ❂ ✵ ❀ ✤ ● ✭✸✮ ❂ ✻ ❀ ✿ ✿ ✿ Read-Hoggar conjecture (1968,1974) The coefficients of the chromatic polynomial ✤ ● ✭ q ✮ form a log-concave sequence for any graph ● , that is, ❛ ✷ ✐ ✕ ❛ ✐ � ✶ ❛ ✐ ✰✶ for all ✐ . June Huh 3 / 26

✶ q ✹ � ✹ q ✸ ✰ ✻ q ✷ � ✸ q ✤ ● ♥ ❡ ✭ q ✮ ❂ ✶ q ✸ � ✷ q ✷ ✰ q ❀ ✤ ● ❂ ❡ ✭ q ✮ ❂ ✤ ● ✭ q ✮ ❂ ✤ ● ♥ ❡ ✭ q ✮ � ✤ ● ❂ ❡ ✭ q ✮ ❂ ✶ q ✹ � ✺ q ✸ ✰ ✽ q ✷ � ✹ q ✿ ✤ ● ✭ q ✮ Example How do we compute the chromatic polynomial? We write ✎ ✎ ✎ ✎ ✎ = - ✎ ✎ ✎ ✎ ✎ ✎ and use ✤ ● ✭ q ✮ ❂ ✤ ● ♥ ❡ ✭ q ✮ � ✤ ● ❂ ❡ ✭ q ✮ ✿ June Huh 4 / 26

Example How do we compute the chromatic polynomial? We write ✎ ✎ ✎ ✎ ✎ = - ✎ ✎ ✎ ✎ ✎ ✎ and use ✤ ● ✭ q ✮ ❂ ✤ ● ♥ ❡ ✭ q ✮ � ✤ ● ❂ ❡ ✭ q ✮ ✿ From the calculation ✶ q ✹ � ✹ q ✸ ✰ ✻ q ✷ � ✸ q ✤ ● ♥ ❡ ✭ q ✮ ❂ ✶ q ✸ � ✷ q ✷ ✰ q ❀ ✤ ● ❂ ❡ ✭ q ✮ ❂ Therefore ✤ ● ✭ q ✮ ❂ ✤ ● ♥ ❡ ✭ q ✮ � ✤ ● ❂ ❡ ✭ q ✮ ❂ ✶ q ✹ � ✺ q ✸ ✰ ✽ q ✷ � ✹ q ✿ This algorithmic description of ✤ ● ✭ q ✮ makes the prediction of the conjecture interesting. June Huh 4 / 26

❢ ✐ ✭ ❆ ✮ ❢ ✐ ✭ ❆ ✮ ❂ ❢ ✐ ✭ ❆ ♥ ✈ ✮ ✰ ❢ ✐ � ✶ ✭ ❆ ❂ ✈ ✮ ✿ For any finite set of vectors ❆ in a vector space over a field, define ❢ ✐ ✭ ❆ ✮ ❂ ✭ number of independent subsets of ❆ with size ✐ ✮ ✿ Example If ❆ is the set of all nonzero vectors in F ✸ ✷ , then ❢ ✵ ❂ ✶ ❀ ❢ ✶ ❂ ✼ ❀ ❢ ✷ ❂ ✷✶ ❀ ❢ ✸ ❂ ✷✽ ✿ June Huh 5 / 26

For any finite set of vectors ❆ in a vector space over a field, define ❢ ✐ ✭ ❆ ✮ ❂ ✭ number of independent subsets of ❆ with size ✐ ✮ ✿ Example If ❆ is the set of all nonzero vectors in F ✸ ✷ , then ❢ ✵ ❂ ✶ ❀ ❢ ✶ ❂ ✼ ❀ ❢ ✷ ❂ ✷✶ ❀ ❢ ✸ ❂ ✷✽ ✿ How do we compute ❢ ✐ ✭ ❆ ✮ ? We use ❢ ✐ ✭ ❆ ✮ ❂ ❢ ✐ ✭ ❆ ♥ ✈ ✮ ✰ ❢ ✐ � ✶ ✭ ❆ ❂ ✈ ✮ ✿ June Huh 5 / 26

Welsh-Mason conjecture (1971,1972) The sequence ❢ ✐ form a log-concave sequence for any finite set of vectors ❆ in any vector space over any field, that is, ❢ ✷ ✐ ✕ ❢ ✐ � ✶ ❢ ✐ ✰✶ for all ✐ . June Huh 6 / 26

▼ ❊ ❊ ❋ ✶ ❋ ✷ ❋ ✶ ❭ ❋ ✷ ❋ ❋ ❊ ♥ ❋ Definition of matroids Whitney, Nakasawa, Birkhoff (1935), Mac Lane (1936), van der Waerden (1937) . . . Let ❊ be a finite set. June Huh 7 / 26

Definition of matroids Whitney, Nakasawa, Birkhoff (1935), Mac Lane (1936), van der Waerden (1937) . . . Let ❊ be a finite set. A matroid ▼ on ❊ is a collection of subsets of ❊ , called (proper) flats , satisfying: (1) If ❋ ✶ and ❋ ✷ are flats, then ❋ ✶ ❭ ❋ ✷ is a flat, (2) If ❋ is a flat, then the flats covering ❋ forms a partition of ❊ ♥ ❋ . June Huh 7 / 26

We write ♥ ✰ ✶ for the size of ▼ , the cardinality of the ground set ❊ . We write r ✰ ✶ for the rank of ▼ , the height of the poset of flats of ▼ . In all interesting cases (except one), r ❁ ♥ . June Huh 8 / 26

❦ ❦ Example A projective space is a set with distinguished subsets, called lines , satisfying: (1) Any two distinct points are in exactly one line. (2) Each line contains more than two points. (3) If ① ❀ ② ❀ ③ ❀ ✇ are distinct points, no three colinear, then ①② intersects ③✇ ❂ ✮ ①③ intersects ②✇ ✿ June Huh 9 / 26

❦ ❦ Example A projective space is a set with distinguished subsets, called lines , satisfying: (1) Any two distinct points are in exactly one line. (2) Each line contains more than two points. (3) If ① ❀ ② ❀ ③ ❀ ✇ are distinct points, no three colinear, then ①② intersects ③✇ ❂ ✮ ①③ intersects ②✇ ✿ A projective space has a structure of flats (subspaces), and this structure is inherited by any of its finite subset, defining a matroid. June Huh 9 / 26

Example A projective space is a set with distinguished subsets, called lines , satisfying: (1) Any two distinct points are in exactly one line. (2) Each line contains more than two points. (3) If ① ❀ ② ❀ ③ ❀ ✇ are distinct points, no three colinear, then ①② intersects ③✇ ❂ ✮ ①③ intersects ②✇ ✿ A projective space has a structure of flats (subspaces), and this structure is inherited by any of its finite subset, defining a matroid. Matroids arising from the projective space over a field ❦ are said to be realizable over ❦ (the idea of “ coordinates ”). June Huh 9 / 26

● ❊ ❊ ▼ ❆ ❦ ❆ ❦ ▼ ❦ Matroids are determined by their independent sets (the idea of “ general position ”), and can be axiomatized in terms of independent sets. June Huh 10 / 26

❆ ❦ ❆ ❦ ▼ ❦ Matroids are determined by their independent sets (the idea of “ general position ”), and can be axiomatized in terms of independent sets. 1. Let ● be a finite graph, and ❊ the set of edges. Call a subset of ❊ independent if it does not contain a circuit. This defines a graphic matroid ▼ . June Huh 10 / 26

Matroids are determined by their independent sets (the idea of “ general position ”), and can be axiomatized in terms of independent sets. 1. Let ● be a finite graph, and ❊ the set of edges. Call a subset of ❊ independent if it does not contain a circuit. This defines a graphic matroid ▼ . 2. Let ❆ a finite subset of a field containing ❦ . Call a subset of ❆ independent if it is algebraically independent over ❦ . This defines a matroid ▼ algebraic over ❦ . June Huh 10 / 26

The Fano matroid is realizable iff char ✭ ❦ ✮ ❂ ✷ . The non-Fano matroid is realizable iff char ✭ ❦ ✮ � ✷ . The non-Pappus matroid is not realizable over any field, but. . . How many matroids are realizable over a field? June Huh 11 / 26

❦ ❂ ❦ ❂ ❀ ❀ 0% of matroids are realizable over a field. In other words, almost all matroids are (conjecturally) not realizable over any field. June Huh 12 / 26

0% of matroids are realizable over a field. In other words, almost all matroids are (conjecturally) not realizable over any field. Testing the realizability of a matroid over a given field is not easy. When ❦ ❂ Q , this is equivalent to Hilbert’s tenth problem over Q . When ❦ ❂ R ❀ C ❀ etc, there are universality theorems on realization spaces. June Huh 12 / 26

One can define the characteristic polynomial of a matroid by the recursion ✤ ▼ ✭ q ✮ ❂ ✤ ▼ ♥ ❡ ✭ q ✮ � ✤ ▼ ❂ ❡ ✭ q ✮ ✿ Rota-Welsh conjecture (1970, 1976) The coefficients of the characteristic polynomial ✤ ▼ ✭ q ✮ form a log-concave sequence for any matroid ▼ , that is, ✖ ✷ ✐ ✕ ✖ ✐ � ✶ ✖ ✐ ✰✶ for all ✐ . This implies the conjecture on ● and the conjecture on ❆ ( Brylawski ). June Huh 13 / 26

✝ ▼ ▼ ✷ A tropical viewpoint A matroid ▼ on ❊ can be viewed as an r -dimensional fan in an ♥ -dimensional space ✝ ▼ ✒ R ❊ ❂ R whose maximal cones correspond to flags of flats ∅ � ❋ ✶ � ❋ ✷ � ✁ ✁ ✁ � ❋ r � ❊ ✿ More precisely, the maximal cones are of the form ❳ e ✐ ✷ R ❊ ❂ R ✿ cone ✭ e ❋ ✶ ❀ ✿ ✿ ✿ ❀ e ❋ r ✮ ❀ e ❋ ❂ ✐ ✷ ❋ June Huh 14 / 26