Toric Algebra Research Articles

Chain algebras of finite distributive lattices

We introduce a family of toric algebras defined by maximal chains of a finite distributive lattice. Applying results on stable set polytopes, we conclude that every such algebra is normal and Cohen–Macaulay, and give an interpretation of its Krull dimension in terms of the combinatorics of the underlying lattice. When the lattice is planar, we show that the corresponding chain algebra is generated by a sortable set of monomials and is isomorphic to a Hibi ring of another finite distributive lattice. As a consequence, it has a defining toric ideal with a quadratic Gröbner basis, and its h-vector counts ascents in certain standard Young tableaux. If instead the lattice has dimension n>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n>2$$\\end{document}, we show that the defining ideal has minimal generators of degree at least n.

Cycle algebras and polytopes of matroids

Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an algebraic and geometric investigation of these polytopes by studying their toric algebras, called cycle algebras, and their defining ideals. Several matroid operations are considered which determine faces of cycle polytopes that belong again to this class of polyhedral objects. As a key technique used in this paper, we study certain minors of given matroids which yield algebra retracts on the level of cycle algebras. In particular, that allows us to use a powerful algebraic machinery. As an application, we study highest possible degrees in minimal homogeneous systems of generators of defining ideals of cycle algebras as well as interesting cases of cut polytopes and Eulerian subgraph polytopes.

State polytopes related to two classes of combinatorial neural codes

Combinatorial neural codes are collections of 0/1 vectors that are used to model the co-firing patterns of a set of place cells in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically drawn on the plane as a Venn diagram-like figure. A sufficient condition to do so is for the code to have a property called 2-inductively pierced. Gross, Obatake, and Youngs recently used toric algebra to show that a code on three neurons is 1-inductively pierced if and only if the toric ideal is trivial or generated by quadratics. No result is known for additional neurons in the same generality, part of the difficulty coming from the large number of codewords possible when additional neurons are used.In this article, we study two infinite classes of combinatorial neural codes in detail. For each code, we explicitly compute its universal Gröbner basis. This is done for the first class by recognizing that the codewords form a Lawrence-type matrix. With the second class, this is done by showing that the matrix is totally unimodular. These computations allow one to compute the state polytopes of the corresponding toric ideals, from which all distinct initial ideals may be computed efficiently. Moreover, we show that the state polytopes are combinatorially equivalent to well-known polytopes: the permutohedron and the stellohedron.

On regularity bounds and linear resolutions of toric algebras of graphs

Let G be a simple graph. We show that if G is connected and R(I (G)) is normal, reg ⁡(R(I (G)))≤α0 (G), where α0 (G) is the vertex cover number of G. As a consequence, for every normal König connected graph G, reg ⁡(R(I (G)))= mat ⁡ (G), the matching number of G. For a gap-free graph G, we give various combinatorial upper bounds for reg ⁡(R(I (G))). As a consequence we give various sufficient conditions for the equality of reg ⁡(R(I (G))) and mat ⁡ (G). Finally we show that if G is a chordal graph such that K[G] has q-linear resolution (q≥4), then K[G] is a hypersurface, which proves the conjecture of Hibi, Matsuda and Tsuchiya [10, Conjecture 0.2] affirmatively for chordal graphs.

Universal Gröbner bases of toric ideals of combinatorial neural codes

In the 1970s, O'Keefe and Dostrovsky discovered that certain neurons, called place cells, in an animal's brain are tied to its location within its arena. A combinatorial neural code is a collection of $0/1$-vectors which encode the patterns of co-firing activity among the place cells. Gross, Obatake, and Youngs have recently used techniques from toric algebra to study when a neural code is $0$- $1$-, or $2$-inductively pierced: a property that allows one to reconstruct a Venn diagram-like planar figure that acts as a geometric schematic for the neural co-firing patterns. This article examines their work closely by focusing on a variety of classes of combinatorial neural codes. In particular, we identify universal Gr\"obner bases of the toric ideal for these codes.

Binomial ideals of domino tilings

In this paper, we consider the set of all domino tilings of a cubiculated region. The primary question we explore is: How can we move from one tiling to another? Tiling spaces can be viewed as spaces of subgraphs of a fixed graph with a fixed degree sequence. Moves to connect such spaces have been explored in algebraic statistics. Thus, we approach this question from an applied algebra viewpoint, making new connections between domino tilings, algebraic statistics, and toric algebra. Using results from toric ideals of graphs, we are able to describe moves that connect the tiling space of a given cubiculated region of any dimension. This is done by studying binomials that arise from two distinct domino tilings of the same region. Additionally, we introduce tiling ideals and flip ideals and use these ideals to restate what it means for a tiling space to be flip connected. Finally, we show that if R is a 2-dimensional simply connected cubiculated region, any binomial arising from two distinct tilings of R can be written in terms of quadratic binomials. As a corollary to our main result, we obtain an alternative proof to the fact that the set of domino tilings of a 2-dimensional simply connected region is connected by flips.

Gorenstein $T$-spread Veronese algebras

Let $S=K[x_1, x_2, \dots, x_n]$ be the polynomial ring in $n$ variables over a field $K$. We fix integers $d$ and $t$. A monomial $x_{i_1}x_{i_2}\cdots x_{i_d}$ with $i_1\leq i_2\leq\dots \leq i_d$ is $t$-spread if $i_j-i_{j-1}\geq t$, for any $2\leq j\leq n$. Let $I_{n,d,t}$ be the ideal generated by all $t$-spread monomials of degree $d$ and let $K[I_{n,d,t}]$ be the toric algebra generated by the monomials $v$ with $v\in G(I_{n,d,t})$. This generalizes the classical (squarefree)Veronese algebras. The aim of this paper is to characterize the algebras $K[I_{n,d,t}]$ which are Gorenstein.

Towards classifying toric degenerations of cubic surfaces

We investigate the class of degenerations of smooth cubic surfaces which are obtained from degenerating their Cox rings to toric algebras. More precisely, we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii bases to determine toric degenerations of Del Pezzo surfaces of degree 4 and who leave the question of classifying these degenerations in the degree 3 case as an open problem. In order to carry out this classification we describe an approach which is closely related to tropical geometry and present partial results in this direction.

Betti numbers of toric algebras of certain bipartite graphs

In this paper, we compute the graded Betti numbers of toric algebras of certain bipartite graphs [Formula: see text]. Also, the Castelnuovo–Mumford regularity, Hilbert series and multiplicity of [Formula: see text] are explicitly determined.

On the Gauss algebra of toric algebras

Let A be a K-subalgebra of the polynomial ring $$S=K[x_1,\ldots ,x_d]$$ of dimension d, generated by finitely many monomials of degree r. Then, the Gauss algebra $${\mathbb {G}}(A)$$ of A is generated by monomials of degree $$(r-1)d$$ in S. We describe the generators and the structure of $${\mathbb {G}}(A)$$, when A is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree 2, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph G with one loop, the embedding dimension of $${\mathbb {G}}(A)$$ is bounded by the complexity of the graph G.

Pattern-avoiding polytopes

Two well-known polytopes whose vertices are indexed by permutations in the symmetric group Sn are the permutohedron Pn and the Birkhoff polytope Bn. We consider polytopes Pn(Π) and Bn(Π), whose vertices correspond to the permutations in Sn avoiding a set of patterns Π. For various choices of Π, we explore the Ehrhart polynomials and h∗-vectors of these polytopes as well as other aspects of their combinatorial structure.For Pn(Π), we consider all subsets Π⊆S3 and are able to provide results in most cases. To illustrate, Pn(123,132) is a Pitman–Stanley polytope, the number of interior lattice points in Pn(132,312) is a derangement number, and the normalized volume of Pn(123,231,312) is the number of trees on n vertices.The polytopes Bn(Π) seem much more difficult to analyze, so we focus on four particular choices of Π. First we show that the Bn(231,321) is exactly the Chan–Robbins–Yuen polytope. Next we prove that for any Π containing {123,312} we have h∗(Bn(Π))=1. Finally, we study Bn(132,312) and B˜n(123), where the tilde indicates that we choose vertices corresponding to alternating permutations avoiding the pattern 123. In both cases we use order complexes of posets and techniques from toric algebra to construct regular, unimodular triangulations of the polytopes. The posets involved turn out to be isomorphic to the lattices of Young diagrams contained in a certain shape, and this permits us to give an exact expression for the normalized volumes of the corresponding polytopes via the hook formula. Finally, Stanley’s theory of (P,ω)-partitions allows us to show that their h∗-vectors are symmetric and unimodal.Various questions and conjectures are presented throughout.

Uniform symbolic topologies in normal toric rings

Given a normal toric algebra R, we compute a uniform integer D=D(R)>0 such that the symbolic power P(DN)⊆PN for all N>0 and all monomial primes P. We compute the multiplier D explicitly in terms of the polyhedral cone data defining R, illustrating the output for Segre–Veronese algebras.

Retracts and algebraic properties of cut algebras

Given a graph G, the cut polytope is the convex hull of its cut vectors. The latter objects are the incidence vectors associated to all cuts of G. Especially motivated by related conjectures of Sturmfels and Sullivant, we study various properties and invariants of the toric algebra of the cut polytope, called its cut algebra. In particular, we characterize those cut algebras which are complete intersections, have linear resolutions or have Castelnuovo–Mumford regularity equal to 2. The key idea of our approach is to consider suitable algebra retracts of cut algebras. Additionally, combinatorial retracts of the graph are defined and investigated, which are special minors whose algebraic properties can be compared in a very pleasant way with the corresponding ones of the original graph. Moreover, we discuss several examples and pose new problems as well.

Modular Curvature and Morita Equivalence

The curvature of the noncommutative torus $T^2_\theta$ ($\theta$ irrational) endowed with a noncommutative conformal metric has been the focus of attention of several recent works. Continuing the approach taken in the paper [A. Connes and H. Moscovici, http://arxiv.org/abs/1110.3500] we extend the study of the curvature to twisted Dirac spectral triples constructed out of Heisenberg bimodules that implement the Morita equivalence of the $C^*$-algebra $A_\theta = C(T^2_\theta)$ with other toric algebras $A_{\theta'}$. In the enlarged context the conformal metric on $T^2_\theta$ is exchanged with an arbitrary Hermitian metric on the Heisenberg $(A_\theta, A_{\theta'})$-bimodule $E'$ for which ${\rm End}_{A_{\theta'}}(E') = A_\theta $. We prove that the Ray-Singer log-determinant of the corresponding Laplacian, viewed as a functional on the space of all Hermitian metrics on $E'$, attains its extremum at the unique Hermitian metric whose corresponding connection has constant curvature. The gradient of the log-determinant functional gives rise to a noncommutative analogue of the Gaussian curvature. The genuinely new outcome of this paper is that the latter is shown to be independent of any Heisenberg bimodule $E'$ such that $A_\theta = {\rm End}_{A_{\theta'}}(E')$, and in this sense it is Morita invariant. To prove the above results we extend Connes' pseudodifferential calculus to Heisenberg modules. The twisted version, which offers more flexibility even in the case of trivial coefficients, could potentially be applied to other problems in the elliptic theory on noncommutative tori. A noteworthy technical feature is that we systematize the computation of the resolvent expansion for elliptic differential operators on noncommutative tori to an extent which makes the (previously employed) computer assistance unnecessary.

Fixed Ideals of Certain Cartier Algebras Over F-split Toric Algebras

In this article, the ideals of an F-split toric algebra that are fixed by certain Cartier algebra are described, extending the result for normal toric algebras in the author's joint work with K. Schwede and W. Zhang.

Goodness of fit for log-linear network models: dynamic Markov bases using hypergraphs

Social networks and other sparse data sets pose significant challenges for statistical inference, since many standard statistical methods for testing model/data fit are not applicable in such settings. Algebraic statistics offers a theoretically justified approach to goodness-of-fit testing that relies on the theory of Markov bases. Most current practices require the computation of the entire basis, which is infeasible in many practical settings. We present a dynamic approach to explore the fiber of a model, which bypasses this issue, and is based on the combinatorics of hypergraphs arising from the toric algebra structure of log-linear models. We demonstrate the approach on the Holland–Leinhardt $$p_1$$ model for random directed graphs that allows for reciprocation effects.

Vanishing ideals over complete multipartite graphs

We study the vanishing ideal of the parametrized algebraic toric set associated to the complete multipartite graph G=Kα1,…,αr over a finite field of order q. We give an explicit family of binomial generators for this lattice ideal, consisting of the generators of the ideal of the torus (referred to as type I generators), a set of quadratic binomials corresponding to the cycles of length 4 in G and which generate the toric algebra ofG (type II generators) and a set of binomials of degree q−1 obtained combinatorially from G (type III generators). Using this explicit family of generators of the ideal, we show that its Castelnuovo–Mumford regularity is equal to max{α1(q−2),…,αr(q−2),⌈(n−1)(q−2)/2⌉}, where n=α1+⋯+αr.

Toric algebra of hypergraphs

The edges of any hypergraph parametrize a monomial algebra called the edge subring of the hypergraph. We study presentation ideals of these edge subrings, and describe their generators in terms of balanced walks on hypergraphs. Our results generalize those for the defining ideals of edge subrings of graphs, which are well-known in the commutative algebra community, and popular in the algebraic statistics community. One of the motivations for studying toric ideals of hypergraphs comes from algebraic statistics, where generators of the toric ideal give a basis for random walks on fibers of the statistical model specified by the hypergraph. Further, understanding the structure of the generators gives insight into the model geometry.

Commutative algebra of statistical ranking

A model for statistical ranking is a family of probability distributions whose states are orderings of a fixed finite set of items. We represent the orderings as maximal chains in a graded poset. The most widely used ranking models are parameterized by rational function in the model parameters, so they define algebraic varieties. We study these varieties from the perspective of combinatorial commutative algebra. One of our models, the Plackett–Luce model, is non-toric. Five others are toric: the Birkhoff model, the ascending model, the Csiszár model, the inversion model, and the Bradley–Terry model. For these models we examine the toric algebra, its lattice polytope, and its Markov basis.

$D$-module structure of local cohomology modules of toric algebras

Let S be a toric algebra over a field K of characteristic 0 and let I be a monomial ideal of S. We show that the local cohomology modules H i I (S) are of finite length over the ring of differential operators D(S;K), generalizing the classical case of a polynomial algebra S. As an application, we compute the characteristic cycles of some local cohomology modules.