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The intersection ring of matroids

Simon Hampe
2017 Journal of combinatorial theory. Series B (Print)  
The product is given by matroid intersection and the additive structure is defined by assigning to each matroid the indicator vector of its chains of flats.  ...  Finally we show that the ring is the cohomology ring of the toric variety of the permutohedron and thus fulfills Poincar\'e duality.  ...  The intersection ring of matroids. The notion of matroid intersection seems to have gotten very little attention from matroid theorists.  ... 
doi:10.1016/j.jctb.2016.08.004 fatcat:dig7ni6ue5bu3gpiorr4sjbznu

Witt rings and matroids

Thomas C. Craven, Zachary A. Kent
2012 Proceedings of the American Mathematical Society  
The study of Witt rings of formally real fields in the algebraic theory of quadratic forms has led to a particularly good understanding of the finitely generated torsion free Witt rings.  ...  In this paper, we work primarily with a somewhat more general class of rings which can be completely characterized by (binary) matroids.  ...  Introduction We begin with a class of quotient rings of integral group rings. These rings, modulo their nilradicals, are the basic objects we wish to classify with matroids.  ... 
doi:10.1090/s0002-9939-2012-11428-5 fatcat:jukndbw57jgvdpiyr4vipfaofi

Oriented Matroids and Combinatorial Neural Codes [article]

Alexander Kunin, Caitlin Lienkaemper, Zvi Rosen
2020 arXiv   pre-print
We adapt the oriented matroid ideal introduced by Novik, Postnikov, and Sturmfels into a functor from the category of oriented matroids to the category of rings; then, we show that the resulting ring maps  ...  naturally to the neural ring of the matroid's neural code.  ...  Acknowledgements Acknowledgements The authors thank Carina Curto, Emanuele Delucchi, Boaz Haberman, Vladimir Itskov, R.  ... 
arXiv:2002.03542v1 fatcat:3ahilydpond3lp3rraxpmjoioe

Page 1131 of Mathematical Reviews Vol. 52, Issue 4 [page]

1976 Mathematical Reviews  
A subset of #" is called (intersection) convex if it is the intersection of #" with a convex set of A(R).  ...  a nonempty intersection if each subcollection of 2" members has a nonempty intersection.  ... 

Cohen–Macaulayness of large powers of Stanley–Reisner ideals

Naoki Terai, Ngo Viet Trung
2012 Advances in Mathematics  
We prove that for m 3, the symbolic power I (m) of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex is a matroid.  ...  Similarly, the ordinary power I m is Cohen-Macaulay for some m 3 if and only if I is a complete intersection.  ...  We call locally a matroid or a complete intersection if the links of at the vertices are matroids or complete intersections, respectively.  ... 
doi:10.1016/j.aim.2011.10.004 fatcat:ruuvqfxberem7ifky4zfurvf4u

Graded Linearity of Stanley–Reisner Ring of Broken Circuit Complexes

Mohammad Reza-Rahmati, Gerardo Flores, Carmelo Antonio Finocchiaro
2022 Journal of Mathematics  
to the decomposition of matroids into the direct sum of uniform matroids.  ...  On the other hand, we provide specific characterizations of graded linear resolution property for the Stanley–Reisner ring of broken circuit complexes and generalize the results of Van Le and Römer, related  ...  commutative rings and the combinatorics of their matroids.  ... 
doi:10.1155/2022/1806967 fatcat:gzpz5cz6gjeudk43nesk3hl23e

On the matroid stratification of Grassmann varieties, specialization of coordinates, and a problem of N. White

Bernd Sturmfels
1989 Advances in Mathematics  
We have the following necessary condition for matroids F whose stratum rF intersects the closure of Tc. for the corresponding maximal ideal in the coordinate ring of the Grassmann variety.  ...  One main new result of their paper is the equivalence of the definition of the matroid stratification given here with the definition of Z,,, as a multi-intersection of all possible Schubert cells with  ... 
doi:10.1016/0001-8708(89)90037-6 fatcat:anutn25ut5cfjd2ebsprgmr6je

Simplicial generation of Chow rings of matroids [article]

Spencer Backman, Christopher Eur, Connor Simpson
2021 arXiv   pre-print
We introduce a presentation of the Chow ring of a matroid by a new set of generators, called "simplicial generators."  ...  Using this combinatorial interpretation, we (i) produce a bijection between a monomial basis of the Chow ring and a relative generalization of Schubert matroids, (ii) recover the Poincar\'e duality property  ...  of the Chow ring of a matroid M .  ... 
arXiv:1905.07114v4 fatcat:qgqusegh55ht5lyi5hnip2lsle

On the Gorensteinness of broken circuit complexes and Orlik--Terao ideals [article]

Le Van Dinh
2014 arXiv   pre-print
It is proved that the broken circuit complex of an ordered matroid is Gorenstein if and only if it is a complete intersection.  ...  Several characterizations for a matroid that admits such an order are then given, with particular interest in the h-vector of broken circuit complexes of the matroid.  ...  The author would like to thank Professor Tim Römer for raising his interest in the topic of the paper and for valuable discussions. He also thanks the referees for useful suggestions.  ... 
arXiv:1305.5795v2 fatcat:tpkgikxqkfh7vbx477k233q6ge

Cohen-Macaulayness of large powers of Stanley-Reisner ideals [article]

Naoki Terai, Ngo Viet Trung
2010 arXiv   pre-print
We prove that for m > 2, the m-th symbolic power of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex is a matroid.  ...  Similarly, the m-th ordinary power is Cohen-Macaulay for some m > 2 if and only if the complex is a complete intersection.  ...  We call ∆ locally a matroid or a complete intersection if the links of ∆ at the vertices are matroids or complete intersections, respectively.  ... 
arXiv:1009.0833v1 fatcat:5gty42addjhaxep6ezvbd4d62y

Broken circuit complexes and hyperplane arrangements [article]

Le Van Dinh, Tim Roemer
2012 arXiv   pre-print
We also show that for an ordered matroid with disjoint minimal broken circuits, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik-Solomon algebra.  ...  We study Stanley-Reisner ideals of broken circuits complexes and characterize those ones admitting a linear resolution or being complete intersections.  ...  It is well-known that when K = C the cohomology ring H . (X , Z) of X depends only on the matroid M(A ): H .  ... 
arXiv:1211.5318v1 fatcat:rws44a3s7vhtbctlk2omgxp4bq

Broken circuit complexes and hyperplane arrangements

Dinh Van Le, Tim Römer
2013 Journal of Algebraic Combinatorics  
We also show that for a matroid with a complete intersection broken circuit complex, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik-Solomon algebra.  ...  As an application, we improve a result of Wilf on upper bounds for the coefficients of the chromatic polynomial of a maximal planar graph.  ...  It is well-known that when K = C the cohomology ring H . (X, Z) of X depends only on the matroid M(A ): H .  ... 
doi:10.1007/s10801-013-0435-z fatcat:r4t2wqgvurfkbm6utt7bgxtwkm

A Generalization to Bases Common to r Binary Matroids A the Weighted Matrix-Tree Theorem in the Case when the Weights are Boolean

Kenneth A. Berman
1984 European journal of combinatorics (Print)  
. , M , be r bina ry matroids of the same rank o n the same set S~{5" 52, ... , 5 m } . Associate with each element s, of S the weight X i where X i E G F(2), the field of the integers mod 2.  ...  Let 9Jl denote the set of bases common to all the se matroids and con sider the pol ynomial f3(x)~Lb 9lI n . h X i' In thi s paper we obtain a formula for f3 (x} in the form of the permanent of an r.d  ...  ring GFix, y) and the only unit in the ring is 1. 7.  ... 
doi:10.1016/s0195-6698(84)80003-7 fatcat:v6jhgkpoknehhjlcryx4r3d6d4

Symbolic powers and matroids

Matteo Varbaro
2010 Proceedings of the American Mathematical Society  
We prove that all the symbolic powers of a Stanley-Reisner ideal I Δ are Cohen-Macaulay if and only if the simplicial complex Δ is a matroid.  ...  the topic of the work.  ...  Acknowledgements The author would like to thank Margherita Barile for valuable information about Corollary 2.9, Aldo Conca for carefully reading the paper, and Le Dinh Nam for useful conversations about  ... 
doi:10.1090/s0002-9939-2010-10685-8 fatcat:n3yepfvakjb3zj634lsqaevgu4

Graded Linearity of Stanley-Reisner Ring of Broken Circuit Complexes [article]

Mohammad Reza Rahmati, Gerardo Flores
2021 arXiv   pre-print
of matroids into the direct sum of uniform matroids.  ...  On the other hand, we provide specific characterizations of graded linear resolution property for the Stanley-Reisner ring of broken circuit complexes and generalize several results of on the decomposition  ...  Our proof is based on the criterias characterizing the complete intersection properties on the cycle matroid of a graph.  ... 
arXiv:2012.14346v4 fatcat:6s6obtvlhfcplbx55pp5rbwgny
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