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Register a new userMorphisms and order ideals of toric posets
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Authors
Matthew Macauley
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Toric posets are cyclic analogues of finite posets. They can be viewed combinatorially as equivalence classes of acyclic orientations generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper we study toric intervals, morphisms, and order ideals, and we provide a connection to cyclic reducibility and conjugacy in Coxeter groups.
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Let $cX$ be a family of subsets of a finite set $E$. A matroid on $E$ is called an $cX$-matroid if each set in $cX$ is a circuit. We consider the problem of determining when there exists a unique maximal $cX$-matroid in the weak order poset of all $cX$-matroids on $E$, and characterizing its rank function when it exists.
The main objects of the present paper are (i) Hibi rings (toric rings arising from order polytopes of posets), (ii) stable set rings (toric rings arising from stable set polytopes of perfect graphs), and (iii) edge rings (toric rings arising from edge polytopes of graphs satisfying the odd cycle condition). The goal of the present paper is to analyze those three toric rings and to discuss their structures in the case where their class groups have small rank. We prove that the class groups of (i), (ii) and (iii) are torsionfree. More precisely, we give descriptions of their class groups. Moreover, we characterize the posets or graphs whose associated toric rings have rank $1$ or $2$. By using those characterizations, we discuss the differences of isomorphic classes of those toric rings with small class groups.
Motivated by generalizing Khovanovs categorification of the Jones polynomial, we study functors $F$ from thin posets $P$ to abelian categories $mathcal{A}$. Such functors $F$ produce cohomology theories $H^*(P,mathcal{A},F)$. We find that CW posets, that is, face posets of regular CW complexes, satisfy conditions making them particularly suitable for the construction of such cohomology theories. We consider a category of tuples $(P,mathcal{A},F,c)$, where $c$ is a certain ${1,-1}$-coloring of the cover relations in $P$, and show the cohomology arising from a tuple $(P,mathcal{A},F,c)$ is functorial, and independent of the coloring $c$ up to natural isomorphism. Such a construction provides a framework for the categorification of a variety of familiar topological/combinatorial invariants: anything expressible as a rank-alternating sum over a thin poset.
To any toric ideal $I_A$, encoded by an integer matrix $A$, we associate a matroid structure called {em the bouquet graph} of $A$ and introduce another toric ideal called {em the bouquet ideal} of $A$. We show how these objects capture the essential combinatorial and algebraic information about $I_A$. Passing from the toric ideal to its bouquet ideal reveals a structure that allows us to classify several cases. For example, on the one end of the spectrum, there are ideals that we call {em stable}, for which bouquets capture the complexity of various generating sets as well as the minimal free resolution. On the other end of the spectrum lie toric ideals whose various bases (e.g., minimal generating sets, Grobner, Graver bases) coincide. Apart from allowing for classification-type results, bouquets provide a new way to construct families of examples of toric ideals with various interesting properties, such as robustness, genericity, and unimodularity. The new bouquet framework can be used to provide a characterization of toric ideals whose Graver basis, the universal Grobner basis, any reduced Grobner basis and any minimal generating set coincide.
We introduce a generalization of the notion of a negligible morphism and study the associated tensor ideals and thick ideals. These ideals are defined by considering deformations of a given monoidal category $mathcal{C}$ over a local ring $R$. If the maximal ideal of $R$ is generated by a single element, we show that any thick ideal of $mathcal{C}$ admits an explicitely given modified trace function. As examples we consider various Deligne categories and the categories of tilting modules for a quantum group at a root of unity and for a semisimple, simply connected algebraic group in prime characteristic. We prove an elementary geometric description of the thick ideals in quantum type A and propose a similar one in the modular case.
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