Research Inspiration from
Mathematical Sciences
Research Institute for Mathematical Sciences
Kyoto University

International Workshop P-positivity in Matroid Theory and Related Topics

October 4--8, 2021

Speakers | Program | Abstracts | RIMS | Weather in Kyoto | Welcome to Kyoto | Japan Travel Guide| Palace Side hotel

O r g a n i z e r s:

A.N. Kirillov, Y. Namikawa, N. Kishimoto


Invited Speakers:

Richard Stanley (University of Miami (USA)), abstract
Nicholas Proudfoot, (University of Oregon, Eugene, USA), abstract
June Huh, (KIAS (S.Korea) and IAS, Princeton, (USA)), abstract
Petter Rr\{a}nd\'{e}n, KTH, Stockholm (Sweden) ), abstract
Tom Braden (University of Massachusetts Amherst.(USA) ), abstract
Avery St. Dizier (ISU, Illinois (USA)), abstract
Karim Alexander Adiprasito (Hebrew U. Jerusalem (Israel) ), abstract
Christopher Eur ( Harvard University,(USA)), abstract
Leonardo Mihalcea (Virginia Tech University,(USA)), abstract
Graham Denham (University of Western Ontario ,(Canada)), abstract
Suho Oh (Texas State University ,(USA)), abstract
Tsuyoshi Miezali, (Waseda U., Tokyo,(Japan) ), abstract
So Okada, (National Institute of Technology(NIT), Oyama College, Oyama(Japan)), abstract
Akihito Wachi,(Hokkaido University of Education (Japan)), abstract
Tomohiro Sasamoto,(Tokyo Institute of Technology (TIT) (Japan)), abstract
A.N. Kirillov,
(RIMS, Kyoto University), abstract

List of Participants

Y. Namikawa
(RIMS, Kyoto University (Japan)),
Norifumi Kishimoto, (RIMS, Kyoto (Japan)),
Soichi Okada (Nagoya U., Nagoya (Japan) ),
Hirishi Naruse, (U. of Yamanashi, Yamanashi (Japan) ),
Satoshi Naito, (Tokyo Institute of Technology(TIT),Tokyo (Japan) ),
Suijie Wang, (Human U., Beijing (China) ),
Chenghong Zhao, (CSU, Beijing (China) ) ,
Ye Liu, (Xian Jiaotong-Liverpool University (China) ),
Akishi Kato, ( University of Tokyo (Japan))
Hideya Watanabe, (Osaka City University (Japan) )
Weili Guo, (Beijing University of Chemical Technology (China))
Yuto Yamamoto, (IBS Center for Geometry and Physics (S. Korea))
Masao Ishikawa,
(Okayama University (Japan))
Kohei Motegi,
(Tokyo University of Marine Science and Technology (Japan))
Atsushi Katsuda,
(Kyushu University (Japan))
Jonathan Leake,
(TU Berliny (Germany))
Mathew H.Y. Xie,
(Tianjin UT (China))
Jacob Matheme,
University of Bonn and MPI (Germany))
Alejandro Morales,
UMass, Amherst (USA))
Masafumi Shimada,
Kyushu University (Japan))
Masatoshi Noumi,
Kobe University (Japan))
Yasuhiro Ohta,
Kobe University (Japan))
Takeshi Ikeda,
Waseda University (Japan))
Koji Hasegawa,
Tohoku University (Japan))
Lorenzo Venturello ,
KTH Royal Unstitute Stockholm (Sweden))
Travis Scrimshaw,
Osaka City University (Japan))
Ahmed Ashraf,
Habib University (Pakistan ))
Victor Reiner,
University of Minnesota (USA ))
Nikolaos Zygouras,
Warwick University (UK))
Benjamin Schröter,
KTH Royal Institute of Technology (Sweden))
Yasuhide Numata,
Shishu University, Nagao (Japan))
Gleb Nenashev,
Brandes University (USA))
Philip B. Zhang,
Tianjin Normal University (China)
Lauren Williams,
Harvard University (USA))
Hiroyuki Ochiai,
Kyushu University (Japan))
Takashi Imamura,
Chiba University (Japan))


P r o g r a m


The Workshop is to be held at RIMS, online
Octobe 4-8 -- lectures according to the schedule below
August 16-21 -- free disscussions

Tentative lectures schedule

time October 4
October 5
October 6
October 7
October 8
9:30 - 10:20 R.Stanley J.Huh N.Proudfoot G.Denham
10:30 - 11:20 T.Braden N.Proudfoot A. Dizier L.Mihalcea T.Sasamoto
11:30 - 12:20 C.Eur C.Eur S. Oh A.N. Kirillov T. Sasamoto
12:40 - 14:10 Lunch Lunch Lunch Lunch Lunch
14:30 - 15:20 A.N.Kirillov, I Discussions K.Adiprasito K.Adiprasito A. Ashraf
15:30 - 16:20 A.N.Kirillov, II S.Okada T.Miezaki A.Wachi
16:30 - 17:20 P.Branden P.Branden Discussions Discussions


A b s t r a c t s


Nicholas Proudfoot, Oregon U., USA
Kazhdan-Lusztig theory of matroids
Abstract: I will give an introduction to Kazhdan-Lusztig polynomials of matroids and Z-polynomials of matroids, with a focus on examples. I will also say a few words about the proof that these polynomials have non-negative coefficients.

Nicholas Proudfoot, Oregon U.USA) ( University of Oregon, Eugene, USA)
Equivariant log concavity, equivariant real rootedness, and equivariant interlacing
Abstract: This talk will be about polynomials whose coefficients are virtual representations of a finite group. For example, given a matroid with symmetry group W, one can define the $,1r|(Bequivariant characteristic polynomial$,1r}(B be taking the coefficients (up to sign) to be the graded pieces of the Orlik-Solomon algebra, regarded as representations of W. We will define what it means for such a polynomial to be log concave or real rooted, or for two such polynomials to interlace. I$,1ry(Bll give lots of conjectures and partial results based on computer calculations.

Avery St. Dizier, ISU, Illinois , USA
Log-Concavity of Littlewood-Richardson Coefficients
Abstract: Schur polynomials and Littlewood-Richardson numbers are classical objects arising in symmetric function theory, representation theory, and the cohomology of the Grassmannian. First, I will give a quick introduction and history. Next, I will motivate and describe new log-concavity properties of the Littlewood-Richardson numbers and Schur polynomials. I will finish by explaining the mechanism behind the log-concavity properties, Brändén and Huh's theory of Lorentzian polynomials.

June Huh (KIAS (S.Korea) and IAS, Princeton, USA
Tropicalizations of Lorentzian polynomials
Abstract: I will give an overview of the theory of Lorentzian polynomials over the field of real Puiseux series. In this setting, one can tropicalize Lorentzian polynomials, and the class of tropicalized Lorentzian polynomials coincides with the class of M-convex functions in the sense of discrete convex analysis. Using the tropical connection, one can produce Lorentzian polynomials from M-convex functions. (Based on joint work with Petter Br,Ad(Bnd,Ai(Bn).

Richard Stanley, University of Miami (USA)


Tom Braden, University of Massachusetts Amherst.(USA)
Top-heaviness for vector configurations and matroids
Abstract: A 1948 theorem of de Bruijn and Erd\'{o}s says that if n points in a projective plane do not lie all on a line, then they determine at least n lines. More generally, Dowling and Wilson conjectured in 1974 that for any finite set of vectors spanning a d-dimensional vector space, the number of k-dimensional spaces that they span is at most the number of (d-k)-dimensional spaces they span, for all k smaller than d/2. In fact, Dowling and Wilson conjectured this inequality holds more generally for matroids, a combinatorial abstraction of incidence geometries which do not have to arise from actual vector configurations. The conjecture for vector configurations was proved in 2017 by Huh and Wang, using the hard Lefschetz Theorem for intersection cohomology applied to a singular algebraic variety associated to the vector configuration. I will discuss their proof, and also more recent joint work with Huh, Matherne, Proudfoot and Wang which proves the conjecture for all matroids, using a combinatorial replacement for intersection cohomology which makes sense even for non-realizable matroids.

Cristopher Eur, Harvard, (USA)
Hodge-Riemann relations in matroid theory
Abstract: ping and using Hodge-Riemann relations in matroid theory. In the first part, we survey the Hodge theory of Chow rings of matroids, discuss its uses, and present a simplified proof of the Hodge-Riemann relations (in degree 1) using the theory of Lorentzian polynomials. In the second part, we discuss tropical Hodge theory and introduce $,1r|(Btautological classes of matroids,$,1r}(B and then, by combining these two notions, we obtain a log-concavity result for the Tutte polynomial of a matroid that generalizes previously established log-concavity properties for matroids. We conclude by noting some remaining or new questions concerning positivity in matroid theory. Joint works with Spencer Backman, Andrew Berget, Connor Simpson, Hunter Spink, and Dennis Tseng.

Karim A.Adiprasito,Hebrew U. Jerusalem, Israel
Combinatorial Weak and Hard Lefschetz theorems and their applications.
Abstract: I will explain a second proof of the Hard Lefschetz property beyond positivity, based on residue formulas for deranged toric varieties, and indicate implications towards problems in combinatorial topology and lattice sheaves on polytopes.

Leonardo MihalceaVirginia Tech ,USA
Lagrangian Geometry of Matroids Positivity and log concavity conjectures in Cotangent Schubert Calculus.
Abstract: In Cotangent Schubert Calculus, the usual Schubert classes are replaced by characteristic classes of Schubert cells, such as the Chern-Schwartz-MacPherson classes in cohomology and motivic Chern classes in K theory. These are pull backs of certain Lagrangian cycles from the cotangent bundle of a flag manifold, and they may be calculated utilizing Demazure-Lusztig operators from the Hecke algebra. In this talk I will discuss several conjectures about positivity and log-concavity of cotangent Schubert classes, involving the transition matrix to the ordinary Schubert classes, and also about the structure constants from the multiplication of the cotangent Schubert classes. This reports on published and ongoing work, mostly with P. Aluffi, J. Schurmann, and C. Su.

Grahem Denham,University of Western Ontario ,Canada
Lagrangian Geometry of Matroids
Abstract: In joint work with Federico Ardila and June Huh, we introduce the conormal fan of a matroid, which is the Lagrangian analogue of the Bergman fan. We use it to give a Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of a matroid. We develop tools for tropical Hodge theory to show that the conormal fan satisfies Poincaré duality, the Hard Lefschetz property, and the Hodge--Riemann relations. Together, these imply conjectures of Brylawski and Dawson about the log-concavity of the h-vectors of the broken circuit complex and independence complex of a matroid.

Suho Oh,Texas State University, USA
Introduction to positroids
Abstract: Abstract : A A positroid is a special case of a realizable matroid that arose from the study of the totally nonnegative part of the Grassmannian by Postnikov. It has been applied to various fields and also has a very rich combinatorial structure. In this introductory talk, I'll give an introduction to positroids focusing on the various combinatorial objects associated with a positroid, and then show some recent developments focusing on the matroidal aspects, including a joint work with Robert Mcalmon and David Xiang. After that we'll go over how to compute the Tutte polynomial (hence the characteristic polynomial, the main theme of this workshop) of a positroid using the Le-diagram.

So Okada National Institute of Technology(NIT), Oyama College, Oyama,Japan
Merged-log-concavity of rational functions and semi-strongly unimodal sequences
Abstract: We discuss the variation of semi-strongly unimodal sequences by a log-concavity of rational functions. For this, we introduce the notion of merged-log-concavity rational functions (we mainly consider some single-width cases in this talk). Loosely speaking, the merged-log-concavity extends Stanley's q-log-concavity of polynomials.We give explicit merged-log-concave rational functions by q-binomial coefficients and convolutions, extending the Cauchy-Binet formula. Then, we discuss the variation of semi-strongly unimodal sequences by critical points. This gives the golden ratio of quantum dilogarithms as a critical point. If time permits, we consider eta products and some conjectures on the merged-log-concavity.

Tsuyoshi Miezali,Waseda U., Tokyo,Japan
"A generalization of the Tutte polynomials"
In this talk, we introduce the concept of the Tutte polynomials of genus g and discuss some of its properties. We note that the Tutte polynomials of genus one are well-known Tutte polynomials. The main result of this talk declares that the Tutte polynomials of genus g are complete matroid invariants. We will also discuss connections between matroids, codes, and lattices and give some open problems.

Petter Rr\{a}nd\'{e}n, KHT, Stockholm, Sweden

Lorentzian polynomials I and II

Abstract: The class of Lorentzian polynomials contains volume polynomials of convex bodies and pro- jective varieties, as well as homogeneous stable polynomials. I will describe the topology and the combinatorics of the space of Lorentzian polynomials, and provide a useful toolbox of linear operators that preserve the Lorentzian property. In the second part I will show how the theory of Lorentzian polynomials can be applied to problems in combinatorics. The talks are based on joint work with June Huh and Jonathan Leak

The Lefschetz property of the coinvariant algebras of complex reflection groups, and some complete intersections
Abstract: A standard graded Artinian algebra $A = A_0 + A_1 + \cdots + A_c$ is said to have the Lefschetz property if $A_i$ is mapped bijectively onto $A_{c-i}$ by multiplying some linear element to the $(c-2i)$th power for any $i < c/2$. A typical example is the coinvariant algebra of the symmetric group, which is a quotient algebra of a polynomial ring by the ideal generated by the elementary symmetric polynomials. It is isomorphic to the cohomology ring of the flag variety, and the Lefschetz property is proved by the Hard Lefschetz Theorem. The aim of this talk is to show the Lefschetz property of coinvariant rings of (most of) complex reflection groups. There are no cohomology rings behind the complex reflection groups in general. So we utilize algebraic theory of Lefschetz property, the classification by Shepherd-Todd and computer.

Tomohiro Sasamoto Tokyo Institute of Tecnology (TIT),
Skew RSK dynamics, I,II
Abstract: In the first part, we first introduce our skew RSK dynamics as iterated skew RSK maps introduced by Sagan and Stanley in the late 1980s. Then we show that this gives a bijection between a pair of semi-standard tableaux (P,Q) and quadruple (V,W,kappa,nu) where V,W are what we call vertically strict tableaux (equivalent to tensor products of KR crystals) and additional data. The relation to Cauchy identity for q-Whittaker polynomials is also explained. In the second part, we introduce a generalization of Greene invariants in our setting and explain ideas of proof by using a novel realization of affine crystal. If time allows we also describe scattering rules of the skew RSK dynamics. Reference [1] Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto, Skew RSK dynamics: Greene invariants, affine crystals and applications to q-Whittaker polynomials, arXiv: 2106.11922 [2] Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto, Identity between restricted Cauchy sums for the q-Whittaker and skew Schur polynomals, arXiv: 2106.11913

A.N.Kirillov, RIMS, Kyoto University, Japan
Rigged Confugurations, past and present,I,II
Abstract: I will give overview of combinatorial aspects of Rigged Configurations (RC), which were introduced by A.N. Kirillov and N.Y. Reshetikhin in the beginning of 80's of the last century, based on analysis the Bethe Ansatz equations associated wit Heisenberg XXX models. Some applications of RC such as Gale-Ryser theorem,parabolic Kostka polynomials, Domino tableaux, KOH,unimodality of q-Gaussian and related polynomials, RSK, Macdonald a(*,*) Conjecture, ..., will be discussed.

A.N.Kirillov, RIMS, Kyoto University, Japan
FK algebras and posutuvuty
Abstract: I will introduce some class of noncommutative quadratic algebras (including Fomin-Kirillov (FK)algebras), and described their connections with cohomology and K-theory of flag varieties, Schubert and Grothendieck polynomials, as well as state some positivity properties and Conjectures related with FK algebras.