Different moment-angle manifolds arising from two polytopes having the same bigraded Betti numbers

Two simple polytopes of dimension 3 having the identical bigraded Betti numbers but non-isomorphic Tor-algebras are presented. These polytopes provide two homotopically different moment-angle manifolds having the same bigraded Betti numbers. These two simple polytopes are the first examples of polytopes that are (toric) cohomologically rigid but not combinatorially rigid.Comment: 9 page, 2 figures, 2 table

Polynomial-Sized Topological Approximations Using The Permutahedron

Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for $n$ points in $\mathbb{R}^d$, we obtain a $O(d)$-approximation with at most $n2^{O(d \log k)}$ simplices of dimension $k$ or lower. In conjunction with dimension reduction techniques, our approach yields a $O(\mathrm{polylog} (n))$-approximation of size $n^{O(1)}$ for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry. Building on the same geometric concept, we also present a lower bound result on the size of an approximate filtration: we construct a point set for which every $(1+\epsilon)$-approximation of the \v{C}ech filtration has to contain $n^{\Omega(\log\log n)}$ features, provided that $\epsilon <\frac{1}{\log^{1+c} n}$ for $c\in(0,1)$.Comment: 24 pages, 1 figur

Marked chain-order polytopes

We introduce in this paper the marked chain-order polytopes associated to a marked poset, generalizing the marked chain polytopes and marked order polytopes by putting them as extremal cases in an Ehrhart equivalent family. Some combinatorial properties of these polytopes are studied. This work is motivated by the framework of PBW degenerations in representation theory of Lie algebras.Comment: 18 pages, title changed, the relation to string polytopes is remove

Specht Polytopes and Specht Matroids

The generators of the classical Specht module satisfy intricate relations. We introduce the Specht matroid, which keeps track of these relations, and the Specht polytope, which also keeps track of convexity relations. We establish basic facts about the Specht polytope, for example, that the symmetric group acts transitively on its vertices and irreducibly on its ambient real vector space. A similar construction builds a matroid and polytope for a tensor product of Specht modules, giving "Kronecker matroids" and "Kronecker polytopes" instead of the usual Kronecker coefficients. We dub this process of upgrading numbers to matroids and polytopes "matroidification," giving two more examples. In the course of describing these objects, we also give an elementary account of the construction of Specht modules different from the standard one. Finally, we provide code to compute with Specht matroids and their Chow rings.Comment: 32 pages, 5 figure

A New Algorithm in Geometry of Numbers

A lattice Delaunay polytope P is called perfect if its Delaunay sphere is the only ellipsoid circumscribed about P. We present a new algorithm for finding perfect Delaunay polytopes. Our method overcomes the major shortcomings of the previously used method. We have implemented and used our algorithm for finding perfect Delaunay polytopes in dimensions 6, 7, 8. Our findings lead to a new conjecture that sheds light on the structure of lattice Delaunay tilings.Comment: 7 pages, 3 figures; Proceedings of ISVD-07, International Symposium on Voronoi diagrams in Science and Engineering held in July of 2007 in Wales, U

Generalized multiplicities of edge ideals

We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show the $j$-multiplicity is multiplicative over the connected components of a hypergraph, and we explicitly relate the $j$-multiplicity of the edge ideal of a properly connected uniform hypergraph to the Hilbert-Samuel multiplicity of its special fiber ring. In addition, we provide general bounds for the generalized multiplicities of the edge ideals and compute these invariants for classes of uniform hypergraphs.Comment: 24 pages, 6 figures. The results of Theorem 4.6 and Theorem 9.2 are now more general. To appear in Journal of Algebraic Combinatoric