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Positivity Problems and Conjectures in Algebraic Combinatorics
 in Mathematics: Frontiers and Perspectives
, 1999
"... Introduction. Algebraic combinatorics is concerned with the interaction between combinatorics and such other branches of mathematics as commutative algebra, algebraic geometry, algebraic topology, and representation theory. Many of the major open problems of algebraic combinatorics are related to p ..."
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Cited by 69 (1 self)
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Introduction. Algebraic combinatorics is concerned with the interaction between combinatorics and such other branches of mathematics as commutative algebra, algebraic geometry, algebraic topology, and representation theory. Many of the major open problems of algebraic combinatorics are related to positivity questions, i.e., showing that certain integers are nonnegative. The significance of positivity to algebraic combinatorics stems from the fact that a nonnegative integer can have both a combinatorial and an algebraic interpretation. The archetypal algebraic interpretation of a nonnegative integer is as the dimension of a vector space. Thus to show that a certain integer m is nonnegative, it suces to nd a vector space Vm of dimension m. Similarly to show that m n, it suces to nd an injective map Vm ! V n or surjective map V n ! Vm . Of course the inequality m n is equivalent to the positivity statement n m 0, while the injectivity of the map ' : Vm ! V n is equivalent to the
Two decompositions in topological combinatorics with applications to matroid complexes
 Trans. Amer. Math. Soc
, 1997
"... Abstract. This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the n ..."
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Cited by 35 (1 self)
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Abstract. This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of Mshellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the ranknumbers of Mshellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex eardecomposition, and, using results of Kalai and Stanley on hvectors of simplicial polytopes, we show that hvectors of pure rankd simplicial complexes that have this property satisfy h0 ≤ h1 ≤ ·· · ≤ h [d/2] and hi ≤ hd−i for 0 ≤ i ≤ [d/2]. We then show that the abstract simplicial complex formed by the collection of independent sets of a matroid (or matroid complex) admits a special type of convex eardecomposition called a PS eardecomposition. This enables us to construct an associated Mshellable poset, whose set of ranknumbers is the hvector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi [17] that the hvector of a matroid complex satisfies the above two sets of inequalities. 1.
gELEMENTS, FINITE BUILDINGS AND HIGHER COHENMACAULAY CONNECTIVITY
, 2005
"... Chari proved that if ∆ is a (d − 1)dimensional simplicial complex with a convex ear decomposition, then h0 ≤ · · · ≤ h ⌊d/2 ⌋ [7]. Nyman and Swartz raised the problem of whether or not the corresponding gvector is an Mvector [18]. This is proved to be true by showing that the set of pairs (ω, ..."
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Cited by 33 (1 self)
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Chari proved that if ∆ is a (d − 1)dimensional simplicial complex with a convex ear decomposition, then h0 ≤ · · · ≤ h ⌊d/2 ⌋ [7]. Nyman and Swartz raised the problem of whether or not the corresponding gvector is an Mvector [18]. This is proved to be true by showing that the set of pairs (ω, Θ), where Θ is a l.s.o.p. for k[∆], the face ring of ∆, and ω is a gelement for k[∆]/Θ, is nonempty whenever the characteristic of k is zero. Finite buildings have a convex ear decomposition. These decompositions point to inequalities on the flag hvector of such spaces similar in spirit to those examined in [18] for order complexes of geometric lattices. This also leads to connections between higher CohenMacaulay connectivity and conditions which insure that h0 < · · · < hi for a predetermined i.
Linear Programming, the Simplex Algorithm and Simple Polytopes
 Math. Programming
, 1997
"... In the first part of the paper we survey some farreaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concerning the simplex algorithm. We describe subexponential randomized pivot ru ..."
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Cited by 32 (1 self)
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In the first part of the paper we survey some farreaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concerning the simplex algorithm. We describe subexponential randomized pivot rules and upper bounds on the diameter of graphs of polytopes. 1 Introduction: A convex polyhedron is the intersection P of a finite number of closed halfspaces in R d . P is a ddimensional polyhedron (briefly, a dpolyhedron) if the points in P affinely span R d . A convex ddimensional polytope (briefly, a dpolytope) is a bounded convex dpolyhedron. Alternatively, a convex dpolytope is the convex hull of a finite set of points which affinely span R d . A (nontrivial) face F of a dpolyhedron P is the intersection of P with a supporting hyperplane. F itself is a polyhedron of some lower dimension. If the dimension of F is k we call F a kface of P . The empty set and P itself are...
CrissCross Methods: A Fresh View on Pivot Algorithms
 Mathematical Programming
, 1997
"... this paper is to present mathematical ideas and ..."
Random Walks, Totally Unimodular Matrices, and a Randomised Dual Simplex Algorithm
, 2001
"... We discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix. We also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron. ..."
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Cited by 21 (3 self)
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We discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix. We also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron.
Sequentially CohenMacaulay modules and local cohomology
 Proceedings of the international colloquium on algebra, arithmetic and geometry, Tata Institute of Fundamental Research
, 2002
"... Let I ⊂ R be a graded ideal in the polynomial ring R = K[x1,..., xn] where K is a field, and fix a term order <. It has been shown in [17] that the Hilbert functions of the local cohomology modules of R/I are bounded by those of R / in(I), where in(I) denotes the initial ideal of I with respect t ..."
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Cited by 16 (5 self)
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Let I ⊂ R be a graded ideal in the polynomial ring R = K[x1,..., xn] where K is a field, and fix a term order <. It has been shown in [17] that the Hilbert functions of the local cohomology modules of R/I are bounded by those of R / in(I), where in(I) denotes the initial ideal of I with respect to <. In this note we study the question
Reverse lexicographic and lexicographic shifting
 J. ALGEBRAIC COMBIN
, 2005
"... A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, ∆lex — an operation that transforms a monomial ideal of S = k[xi: i ∈ N] that is finitely generated in each degree into a squarefree strongly stable id ..."
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Cited by 13 (2 self)
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A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, ∆lex — an operation that transforms a monomial ideal of S = k[xi: i ∈ N] that is finitely generated in each degree into a squarefree strongly stable ideal — is defined and studied. It is proved that (in contrast to the reverse lexicographic case) a squarefree strongly stable ideal I ⊂ S is fixed by lexicographic shifting if and only if I is a universal squarefree lexsegment ideal (abbreviated USLI) of S. Moreover, in the case when I is finitely generated and is not a USLI, it is verified that all the ideals in the sequence {∆i lex (I)} ∞ i=0 are distinct. The limit ideal ∆(I) = limi→ ∞ ∆i lex (I) is well defined and is a USLI that depends only on a certain analog of the Hilbert function of I.
Algebraic shifting and basic constructions on simplicial complexes
, 2004
"... We try to understand the behavior of algebraic shifting with respect to some basic constructions on simplicial complexes, such as union, coning, and (more generally) join. In particular, for the disjoint union of simplicial complexes we prove ∆(K ˙∪L) = ∆(∆(K) ˙∪∆(L)) (conjectured by Kalai [2]), an ..."
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Cited by 12 (2 self)
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We try to understand the behavior of algebraic shifting with respect to some basic constructions on simplicial complexes, such as union, coning, and (more generally) join. In particular, for the disjoint union of simplicial complexes we prove ∆(K ˙∪L) = ∆(∆(K) ˙∪∆(L)) (conjectured by Kalai [2]), and for the join we give an example of simplicial complexes K and L for which ∆(K ∗ L) ̸ = ∆(∆(K) ∗ ∆(L)) (disproving a conjecture by Kalai [2]), where ∆ denotes the (exterior) algebraic shifting operator. We develop a ’homological ’ point of view on algebraic shifting which is used throughout this work. 1