Pub Date : 2018-12-08DOI: 10.1142/9789811200489_0003
P. Alexandersson, Elie Alhajjar
Demazure characters, also known as key polynomials, generalize the classical Schur polynomials. In particular, when all variables are set equal to $1$, these polynomials count the number of integer points in a certain class of Gelfand--Tsetlin polytopes. This property highlights the interaction between the corresponding polyhedral and combinatorial structures via Ehrhart theory. In this paper, we give an overview of results concerning the interplay between the geometry of Gelfand-Tsetlin polytopes and their Ehrhart polynomials. Motivated by strong computer evidence, we propose several conjectures about the non-negativity of the coefficients of such polynomials.
{"title":"Ehrhart positivity and Demazure characters","authors":"P. Alexandersson, Elie Alhajjar","doi":"10.1142/9789811200489_0003","DOIUrl":"https://doi.org/10.1142/9789811200489_0003","url":null,"abstract":"Demazure characters, also known as key polynomials, generalize the classical Schur polynomials. In particular, when all variables are set equal to $1$, these polynomials count the number of integer points in a certain class of Gelfand--Tsetlin polytopes. This property highlights the interaction between the corresponding polyhedral and combinatorial structures via Ehrhart theory. In this paper, we give an overview of results concerning the interplay between the geometry of Gelfand-Tsetlin polytopes and their Ehrhart polynomials. Motivated by strong computer evidence, we propose several conjectures about the non-negativity of the coefficients of such polynomials.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128205830","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-09-26DOI: 10.1142/9789811200489_0026
Y. Suyama
This article gives an overview of toric Fano and toric weak Fano varieties associated to graphs and building sets. We also study some properties of such toric Fano varieties and discuss related topics.
{"title":"Notes on toric Fano varieties associated to building sets","authors":"Y. Suyama","doi":"10.1142/9789811200489_0026","DOIUrl":"https://doi.org/10.1142/9789811200489_0026","url":null,"abstract":"This article gives an overview of toric Fano and toric weak Fano varieties associated to graphs and building sets. We also study some properties of such toric Fano varieties and discuss related topics.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116657049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-09-24DOI: 10.1142/9789811200489_0020
Jorge Alberto Olarte, Marta Panizzut, Benjamin Schroter
We study the fan structure of Dressians $Dr(d,n)$ and local Dressians $Dr(cM)$ for a given matroid $cM$. In particular we show that the fan structure on $Dr(cM)$ given by the three term Pl"ucker relations coincides with the structure as a subfan of the secondary fan of the matroid polytope $P(cM)$. As a corollary, we have that a matroid subdivision is determined by its 3-dimensional skeleton. We also prove that the Dressian of the sum of two matroids is isomorphic to the product of the Dressians of the matroids. Finally we focus on indecomposable matroids. We show that binary matroids are indecomposable, and we provide a non-binary indecomposable matroid as a counterexample for the converse.
{"title":"On local Dressians of matroids","authors":"Jorge Alberto Olarte, Marta Panizzut, Benjamin Schroter","doi":"10.1142/9789811200489_0020","DOIUrl":"https://doi.org/10.1142/9789811200489_0020","url":null,"abstract":"We study the fan structure of Dressians $Dr(d,n)$ and local Dressians $Dr(cM)$ for a given matroid $cM$. In particular we show that the fan structure on $Dr(cM)$ given by the three term Pl\"ucker relations coincides with the structure as a subfan of the secondary fan of the matroid polytope $P(cM)$. As a corollary, we have that a matroid subdivision is determined by its 3-dimensional skeleton. We also prove that the Dressian of the sum of two matroids is isomorphic to the product of the Dressians of the matroids. Finally we focus on indecomposable matroids. We show that binary matroids are indecomposable, and we provide a non-binary indecomposable matroid as a counterexample for the converse.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127667616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-09-18DOI: 10.1142/9789811200489_0028
Hailun Zheng
We give a survey on the recent results and problems on the face enumeration of flag complexes and flag simplicial spheres, with an emphasis on the characterization of face vectors of flag complexes, several lower-bound type of conjectures including the Charney-Davis conjecture and Gal's conjecture, and the upper bound conjecture for flag spheres and pseudomanifolds.
{"title":"Face enumeration on flag complexes and flag spheres","authors":"Hailun Zheng","doi":"10.1142/9789811200489_0028","DOIUrl":"https://doi.org/10.1142/9789811200489_0028","url":null,"abstract":"We give a survey on the recent results and problems on the face enumeration of flag complexes and flag simplicial spheres, with an emphasis on the characterization of face vectors of flag complexes, several lower-bound type of conjectures including the Charney-Davis conjecture and Gal's conjecture, and the upper bound conjecture for flag spheres and pseudomanifolds.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115712617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-09-12DOI: 10.1142/9789811200489_0022
Irem Portakal
It has been shown by Hacking and Prokhorov that if the projective surface X with quotient singularities and self-intersection number 9 has a smoothing to the projective plane, then X is the general fiber of a Q-Gorenstein deformation of the weighted projective plane with weights giving solutions to the Markov equation. This result has been understood and generalized by combinatorial mutations of Fano triangles by Akhtar, Coates, Galkin, and Kasprzyk. In this note, we study this result by utilizing polarized T-varieties and describe the associated deformation explicitly in terms of certain Minkowski summands of so-called divisorial polytopes.
{"title":"A note on deformations and mutations of fake weighted projective planes","authors":"Irem Portakal","doi":"10.1142/9789811200489_0022","DOIUrl":"https://doi.org/10.1142/9789811200489_0022","url":null,"abstract":"It has been shown by Hacking and Prokhorov that if the projective surface X with quotient singularities and self-intersection number 9 has a smoothing to the projective plane, then X is the general fiber of a Q-Gorenstein deformation of the weighted projective plane with weights giving solutions to the Markov equation. This result has been understood and generalized by combinatorial mutations of Fano triangles by Akhtar, Coates, Galkin, and Kasprzyk. In this note, we study this result by utilizing polarized T-varieties and describe the associated deformation explicitly in terms of certain Minkowski summands of so-called divisorial polytopes.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123837553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-09-05DOI: 10.1142/9789811200489_0015
Lukas Katthan
The Lecture Hall cone is a simplicial cone whose lattice points naturally correspond to Lecture Hall partitions. The celebrated Lecture Hall Theorem of Bousquet-Melou and Eriksson states that a particular specialization of its multivariate Ehrhart series factors in a very nice and unexpected way. Over the years, several proofs of this result have been found, but it is still not considered to be well-understood from a geometric perspective. In this note we propose two conjectures which aim at clarifying this result. Our main conjecture is that the Ehrhart ring of the Lecture Hall cone is actually an initial subalgebra $A_n$ of a certain subalgebra of a polynomial ring, which is itself isomorphic to a polynomial ring. As passing to initial subalgebras does not affect the Hilbert function, this explains the observed factorization. We give a recursive definition of certain Laurent polynomials, which generate the algebra $A_n$. Our second conjecture is that these Laurent polynomials are in fact polynomials. We computationally verified that both conjectures hold for Lecture Hall partitions of length at most 12.
{"title":"The Lecture Hall cone as a toric deformation","authors":"Lukas Katthan","doi":"10.1142/9789811200489_0015","DOIUrl":"https://doi.org/10.1142/9789811200489_0015","url":null,"abstract":"The Lecture Hall cone is a simplicial cone whose lattice points naturally correspond to Lecture Hall partitions. The celebrated Lecture Hall Theorem of Bousquet-Melou and Eriksson states that a particular specialization of its multivariate Ehrhart series factors in a very nice and unexpected way. Over the years, several proofs of this result have been found, but it is still not considered to be well-understood from a geometric perspective. In this note we propose two conjectures which aim at clarifying this result. Our main conjecture is that the Ehrhart ring of the Lecture Hall cone is actually an initial subalgebra $A_n$ of a certain subalgebra of a polynomial ring, which is itself isomorphic to a polynomial ring. As passing to initial subalgebras does not affect the Hilbert function, this explains the observed factorization. We give a recursive definition of certain Laurent polynomials, which generate the algebra $A_n$. Our second conjecture is that these Laurent polynomials are in fact polynomials. We computationally verified that both conjectures hold for Lecture Hall partitions of length at most 12.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126410447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-20DOI: 10.1142/9789811200489_0008
D. Cavey
We determine restrictions on the singularity content of a Fano polygon, or equivalently of certain orbifold del Pezzo surfaces. We establish bounds on the maximum number of 1/R(1,1) singularities in the basket of residual singularities. In particular, there are no Fano polygons without T-singularities and with a basket given by (i) {k x 1/R(1,1)} where k is a positive integer and R>4, or (ii) {1/R1(1,1), 1/R2(1,1), 1/R3(1,1)}.
我们确定了Fano多边形奇异性含量的限制条件，或等价于某些轨道del Pezzo曲面。我们建立了残差奇点篮子中1/R(1,1)个奇点的最大数目的界。特别地，不存在不存在t奇点的Fano多边形，并且不存在由(i) {k x 1/R(1,1)}给出的篮子，其中k是正整数且R>4，或者(ii) {1/R1(1,1)， 1/R2(1,1)， 1/R3(1,1)}给出的篮子。
{"title":"Restrictions on the singularity content of a Fano polygon","authors":"D. Cavey","doi":"10.1142/9789811200489_0008","DOIUrl":"https://doi.org/10.1142/9789811200489_0008","url":null,"abstract":"We determine restrictions on the singularity content of a Fano polygon, or equivalently of certain orbifold del Pezzo surfaces. We establish bounds on the maximum number of 1/R(1,1) singularities in the basket of residual singularities. In particular, there are no Fano polygons without T-singularities and with a basket given by (i) {k x 1/R(1,1)} where k is a positive integer and R>4, or (ii) {1/R1(1,1), 1/R2(1,1), 1/R3(1,1)}.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126689057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-18DOI: 10.1142/9789811200489_0021
McCabe Olsen
Lecture hall partitions are a fundamental combinatorial structure which have been studied extensively over the past two decades. These objects have produced new results, as well as reinterpretations and generalizations of classicial results, which are of interest in combinatorial number theory, enumerative combinatorics, and convex geometry. In a recent survey of Savage cite{Savage-LHP-Survey}, a wide variety of these results are nicely presented. However, since the publication of this survey, there have been many new developments related to the polyhedral geometry and Ehrhart theory arising from lecture hall partitions. Subsequently, in this survey article, we focus exclusively on the polyhedral geometric results in the theory of lecture hall partitions in an effort to showcase these new developments. In particular, we highlight results on lecture hall cones, lecture hall simplices, and lecture hall order polytopes. We conclude with an extensive list of open problems and conjectures in this area.
{"title":"Polyhedral geometry for lecture hall partitions","authors":"McCabe Olsen","doi":"10.1142/9789811200489_0021","DOIUrl":"https://doi.org/10.1142/9789811200489_0021","url":null,"abstract":"Lecture hall partitions are a fundamental combinatorial structure which have been studied extensively over the past two decades. These objects have produced new results, as well as reinterpretations and generalizations of classicial results, which are of interest in combinatorial number theory, enumerative combinatorics, and convex geometry. In a recent survey of Savage cite{Savage-LHP-Survey}, a wide variety of these results are nicely presented. However, since the publication of this survey, there have been many new developments related to the polyhedral geometry and Ehrhart theory arising from lecture hall partitions. Subsequently, in this survey article, we focus exclusively on the polyhedral geometric results in the theory of lecture hall partitions in an effort to showcase these new developments. In particular, we highlight results on lecture hall cones, lecture hall simplices, and lecture hall order polytopes. We conclude with an extensive list of open problems and conjectures in this area.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125519935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-15DOI: 10.1142/9789811200489_0006
Benjamin Braun, Andr'es R. Vindas-Mel'endez
Zonotopes are a rich and fascinating family of polytopes, with connections to many areas of mathematics. In this article we provide a brief survey of classical and recent results related to lattice zonotopes. Our emphasis is on connections to combinatorics, both in the sense of enumeration (e.g. Ehrhart theory) and combinatorial structures (e.g. graphs and permutations).
{"title":"A brief survey on lattice zonotopes","authors":"Benjamin Braun, Andr'es R. Vindas-Mel'endez","doi":"10.1142/9789811200489_0006","DOIUrl":"https://doi.org/10.1142/9789811200489_0006","url":null,"abstract":"Zonotopes are a rich and fascinating family of polytopes, with connections to many areas of mathematics. In this article we provide a brief survey of classical and recent results related to lattice zonotopes. Our emphasis is on connections to combinatorics, both in the sense of enumeration (e.g. Ehrhart theory) and combinatorial structures (e.g. graphs and permutations).","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133160288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-07-22DOI: 10.1142/9789811200489_0024
Liam Solus
There is currently a growing interest in understanding which lattice simplices have unimodal local $h^ast$-polynomials (sometimes called box polynomials); specifically in light of their potential applications to unimodality questions for Ehrhart $h^ast$-polynomials. In this note, we compute a general form for the local $h^ast$-polynomial of a well-studied family of lattice simplices whose associated toric varieties are weighted projective spaces. We then apply this formula to prove that certain such lattice simplices, whose combinatorics are naturally encoded using common systems of numeration, all have real-rooted, and thus unimodal, local $h^ast$-polynomials. As a consequence, we discover a new restricted Eulerian polynomial that is real-rooted, symmetric, and admits intriguing number theoretic properties.
{"title":"Local h*-polynomials of some weighted projective spaces","authors":"Liam Solus","doi":"10.1142/9789811200489_0024","DOIUrl":"https://doi.org/10.1142/9789811200489_0024","url":null,"abstract":"There is currently a growing interest in understanding which lattice simplices have unimodal local $h^ast$-polynomials (sometimes called box polynomials); specifically in light of their potential applications to unimodality questions for Ehrhart $h^ast$-polynomials. In this note, we compute a general form for the local $h^ast$-polynomial of a well-studied family of lattice simplices whose associated toric varieties are weighted projective spaces. We then apply this formula to prove that certain such lattice simplices, whose combinatorics are naturally encoded using common systems of numeration, all have real-rooted, and thus unimodal, local $h^ast$-polynomials. As a consequence, we discover a new restricted Eulerian polynomial that is real-rooted, symmetric, and admits intriguing number theoretic properties.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133309456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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