Gosset’s figure in a Clifford algebra

This note describes a way to realize a “projective” version of Gosset’s 240-vertex semiregular polytope 421 using the Clifford algebra Cl(8) generated by an 8-dimensional vector space equipped with a non-degenerate quadratic form. The 120 vertices of this projective Gosset figure are also seen to coincide with a particular basis for the Lie algebra      

Bispectral quantum Knizhnik�CZamolodchikov equations for arbitrary root systems

The bispectral quantum Knizhnik–Zamolodchikov (BqKZ) equation corresponding to the affine Hecke algebra H of type A _N-1 is a consistent system of q-difference equations which in some sense contains two families of Cherednik’s quantum affine Knizhnik–Zamolodchikov equations for meromorphic functions with values in principal series representations of H. In this paper, we extend this construction of BqKZ to the case where H is the affine Hecke algebra associated with an arbitrary irreducible reduced root system. We construct explicit solutions of BqKZ and describe its correspondence to a bispectral problem involving Macdonald’s q-difference operators.     

Homology of the double and the triple loop spaces¶of E6, E7, and E8

We study the mod p homology of the double and the triple loop spaces of exceptional Lie groups E ₆, E ₇, and E ₈ through the Eilenberg–Moore spectral sequence and the Serre spectral sequence using homology operations. The Bockstein actions on them are also determined. As a result, the Eilenberg–Moore spectral sequences of the path loop fibrations converging to H _*(Ω² G;? _p ) and H _*(Ω³ G;? _p ) collapse at the E ²-term for any compact simple Lie group G. Received: 11 November 1999     

König–Egerváry Graphs are Non-Edmonds

König–Egerváry graphs are those whose maximum matchings are equicardinal to their minimum-order coverings by vertices. Edmonds (J Res Nat Bur Standards Sect B 69B:125–130, 1965) characterized the perfect matching polytope of a graph G = (V, E) as the set of nonnegative vectors ${{\bf{x}}\in\mathbb R^E}K?nig–Egerváry graphs are those whose maximum matchings are equicardinal to their minimum-order coverings by vertices. Edmonds (J Res Nat Bur Standards Sect B 69B:125–130, 1965) characterized the perfect matching polytope of a graph G = (V, E) as the set of nonnegative vectors x ? \mathbb R^E{{\bf{x}}\in\mathbb R^E} satisfying two families of constraints: ‘vertex saturation’ and ‘blossom’. Graphs for which the latter constraints are implied by the former are termed non-Edmonds. This note presents two proofs—one combinatorial, one algorithmic—of its title’s assertion. Neither proof relies on the characterization of non-Edmonds graphs due to de Carvalho et al. (J Combin Theory Ser B 92:319–324, 2004).     

Asymmetry of Convex Polytopes and Vertex Index of Symmetric Convex Bodies

In (Gluskin, Litvak in Geom. Dedicate 90:45–48, [2002]) it was shown that a polytope with few vertices is far from being symmetric in the Banach–Mazur distance. More precisely, it was shown that Banach–Mazur distance between such a polytope and any symmetric convex body is large. In this note we introduce a new, averaging-type parameter to measure the asymmetry of polytopes. It turns out that, surprisingly, this new parameter is still very large, in fact it satisfies the same lower bound as the Banach–Mazur distance. In a sense it shows the following phenomenon: if a convex polytope with small number of vertices is as close to a symmetric body as it can be, then most of its vertices are as bad as the worst one. We apply our results to provide a lower estimate on the vertex index of a symmetric convex body, which was recently introduced in (Bezdek, Litvak in Adv. Math. 215:626–641, [2007]). Furthermore, we give the affirmative answer to a conjecture by Bezdek (Period. Math. Hung. 53:59–69, [2006]) on the quantitative illumination problem.     

On the inductive McKay condition in the defining characteristic

This note is concerned with the McKay conjecture in the representation theory of finite groups. Recently, Isaacs–Malle–Navarro have shown that, in order to prove this conjecture in general, it is sufficient to establish certain properties of all finite simple groups. In this note, we develop some new methods for dealing with these properties for finite simple groups of Lie type in the defining characteristic case. We apply these methods to show that the Suzuki and Ree groups, G ₂(q), F ₄(q) and E ₈(q) have the required properties.     

A counterexample to the dominating set conjecture

The metric polytope met _n is the polyhedron associated with all semimetrics on n nodes and defined by the triangle inequalities x _ij − x _ik − x _jk ≤ 0 and x _ij + x _ik + x _jk ≤ 2 for all triples i, j, k of {1,..., n}. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of the metric polytope is adjacent to some integral vertex. The conjecture holds for n ≤ 8 and, in particular, for the 1,550,825,600 vertices of met₈. While the overwhelming majority of the known vertices of met₉ satisfy the conjecture, we exhibit a fractional vertex not adjacent to any integral vertex.     

The automorphism group of a Banach principal bundle with {1}-structure

A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach–Lie group acting smoothly on P provided the Lie algebra of infinitesimal automorphisms consists of complete vector fields. As a consequence we show that the automorphism group of a connected geodesically complete affine Banach manifold M can be turned into a Banach–Lie group acting smoothly on M.     

Simple vertices of maximal minor polytopes

Themaximal minor polytope Π _{m, n} is the Newton polytope of the product of all maximal minors of anm×n matrix of indeterminates. The family of polytopes {Π _{m, n} } interpolates between the symmetric transportation polytope (form=n−1) and the permutohedron (form=2). Both transportation polytope and the permutohedron aresimple polytopes but in general Π _{m, n} is not simple. The main result of this paper is an explicit construction of a class of simple vertices of Π _{m, n} for generalm andn. We call themvertices of diagonal type. For every such vertexv we explicitly describe all the edges and facets of Π _{m, n} which containv. Simple vertices of Π _{m, n} have an interesting algebro-geometric application: they correspond tononsingular extreme toric degenerations of the determinantal variety ofm×n matrices not of full rank. Andrei Zelevinsky was partially supported by the NSF under Grant DMS-9104867.     

Equipartite polytopes

A polytope P with 2n vertices is called equipartite if for any partition of its vertex set into two equal-size sets V ₁ and V ₂, there is an isometry of the polytope P that maps V ₁ onto V ₂. We prove that an equipartite polytope in ℝ ^d can have at most 2d+2 vertices. We show that this bound is sharp and identify all known equipartite polytopes in ℝ ^d . We conjecture that the list is complete.     

Score Lists in Tripartite Hypertournaments

Given non-negative integers l, m, n, α, β and γ with l ≥ α ≥ 1, m ≥ β ≥ 1 and n ≥ γ ≥ 1, an [α,β,γ]-tripartite hypertournament on l + m + n vertices is a four tuple (U, V, W, E), where U, V and W are three sets of vertices with |U| = l , |V| = m and |W| = n, and E is a set of (α + β + γ)-tuples of vertices, called arcs, with exactly α vertices from U, exactly β vertices from V,and exactly γ vertices from W, such that any subset U₁∪ V₁∪ W₁ of U∪ V∪ W, E contains exactly one of the (α + β + γ)! (α + β + γ) − tuples whose entries belong to U₁∪ V₁∪ W₁. We obtain necessary and sufficient conditions for three lists of non-negative integers in non-decreasing order to be the losing score lists or score lists of some [α, β, γ]-tripartite hypertournament. Supported by National Science Foundation of China (No.10501021).     

Kac–Moody $${\widetilde{E}_k}$$-bundles over elliptic curves and del Pezzo surfaces with singularities of type A

Given an elliptic curve Σ, flat E _k -bundles over Σ are in one-to-one correspondence with smooth del Pezzo surfaces of degree 9 − k containing Σ as an anti-canonical curve. This correspondence was generalized to Lie groups of any type. In this article, we show that there is a similar correspondence between del Pezzo surfaces of degree 0 with an A _d -singularity containing Σ as an anti-canonical curve and Kac–Moody [(E)\tilde]_k{\widetilde{E}_{k}}-bundles over Σ with k = 8 − d. In the degenerate case where surfaces are rational elliptic surfaces, the corresponding [(E)\tilde]_k{\widetilde{E}_k}-bundles over Σ can be reduced to E _k -bundles.     

How many cyclic subpolytopes can a non-cyclic polytope have?

Acyclic d-polytope is ad-polytope that is combinatorially equivalent to a polytope whose vertices lie on the moment curve {(t, t ², …,t ^d):t∈R}. Every subpolytope of an even-dimensional cyclic polytope is again cyclic. We show that a polytope [or neighborly polytope] withv vertices that is not cyclic has at mostd+1 [respectivelyd]d-dimensional cyclic subpolytopes withv−1 vertices, providedd is even andv≧d+5.     

Semidefinite bounds for the stability number of a graph via sums of squares of polynomials

Lovász and Schrijver (SIAM J. Optim. 1:166–190, 1991) have constructed semidefinite relaxations for the stable set polytope of a graph G = (V,E) by a sequence of lift-and-project operations; their procedure finds the stable set polytope in at most α(G) steps, where α(G) is the stability number of G. Two other hierarchies of semidefinite bounds for the stability number have been proposed by Lasserre (SIAM J. Optim. 11:796–817, 2001; Lecture Notes in Computer Science, Springer, Berlin Heidelberg New York, pp 293–303, 2001) and by de Klerk and Pasechnik (SIAM J. Optim. 12:875–892), which are based on relaxing nonnegativity of a polynomial by requiring the existence of a sum of squares decomposition. The hierarchy of Lasserre is known to converge in α(G) steps as it refines the hierarchy of Lovász and Schrijver, and de Klerk and Pasechnik conjecture that their hierarchy also finds the stability number after α(G) steps. We prove this conjecture for graphs with stability number at most 8 and we show that the hierarchy of Lasserre refines the hierarchy of de Klerk and Pasechnik.      

Totally Splittable Polytopes

A split of a polytope is a (necessarily regular) subdivision with exactly two maximal cells. A polytope is totally splittable if each triangulation (without additional vertices) is a common refinement of splits. This paper establishes a complete classification of the totally splittable polytopes.     

Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces

A closed, convex and bounded setP in a Banach spaceE is called a polytope if every finite-dimensional section ofP is a polytope. A Banach spaceE is called polyhedral ifE has an equivalent norm such that its unit ball is a polytope. We prove here:

Facets with fixed defect of the stable set polytope

The stable set polytope of a graph is the convex hull of the 0-1 vectors corresponding to stable sets of vertices. To any nontrivial facet ∑ _v∈V a(v)x _v ≤b of this polytope we associate an integer δ, called the defect of the facet, by δ=∑ _v∈V a(v)-2b. We show that for any fixed δ there is a finite collection of graphs (called “basis”) such that any graph with a facet of defect δ contains a subgraph which can be obtained from one of the graphs in the basis by a simple subdivision operation. Received: September 28, 1998 / Accepted: February 24, 2000?Published online April 20, 2000     

Lattice polytopes,Hecke operators,and the Ehrhart polynomial

Let P be a simple lattice polytope. We define an action of the Hecke operators on E(P), the Ehrhart polynomial of P, and describe their effect on the coefficients of E(P). We also describe how the Brion–Vergne formula for E(P) transforms under the Hecke operators for nonsingular lattice polytopes P.      

The Frattini Subalgebra of Restricted Lie Superalgebras

In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra （L, [p]）. We show first that if L = A1 ＋ A2 ＋… ＋An, then Фp（L） = Фp（A1） ＋Фp（A2） ＋…＋Фp（An）, where each Ai is a p-ideal of L. We then obtain two results： F（L） = Ф（L） = J（L） = L if and only if L is nilpotent; Fp（L） and F（L） are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.