On the adjacency algebras of near hexagons with an order

Suppose is a finite near hexagon of order (s, t) having v points. For every , let denote the adjacency matrix of the graph defined on the points by the distance i relation. We perform a study of the real algebra generated by the 's, and take a closer look to the structure of these algebras for all known examples of . Among other things, we show that a certain number (which is a function of s, t and v) must be integral. This allows us to exclude certain near hexagons whose (non)existence was already open for about 15 years. In the special case , we also show that the embedding rank of the near hexagon is at least the number , and that the near hexagon has non-full projective dimensions with vector dimension equal to d(S)

Cluster adjacency beyond MHV

We explore further the notion of cluster adjacency, focussing on non-MHV amplitudes. We extend the notion of adjacency to the BCFW decomposition of tree-level amplitudes. Adjacency controls the appearance of poles, both physical and spurious, in individual BCFW terms. We then discuss how this notion of adjacency is connected to the adjacency already observed at the level of symbols of scattering amplitudes which controls the appearance of branch cut singularities. Poles and symbols become intertwined by cluster adjacency and we discuss the relation of this property to the $\bar{Q}$-equation which imposes constraints on the derivatives of the transcendental functions appearing in loop amplitudes.Comment: 51 pages, 25 figures, 4 table

The Steinmann Cluster Bootstrap for N=4 Super Yang-Mills Amplitudes

We review the bootstrap method for constructing six- and seven-particle amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory, by exploiting their analytic structure. We focus on two recently discovered properties which greatly simplify this construction at symbol and function level, respectively: the extended Steinmann relations, or equivalently cluster adjacency, and the coaction principle. We then demonstrate their power in determining the six-particle amplitude through six and seven loops in the NMHV and MHV sectors respectively, as well as the symbol of the NMHV seven-particle amplitude to four loops.Comment: 36 pages, 4 figures, 5 tables, 1 ancillary file. Contribution to the proceedings of the Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" (CORFU2019), 31 August - 25 September 2019, Corfu, Greec

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

On tetrahedrally closed line systems and a generalization of the Haemers-Roos inequality

We generalize the well-known Haemers-Roos inequality for generalized hexagons of order (s, t) to arbitrary near hexagons S with an order. The proof is based on the fact that a certain matrix associated with S is idempotent. The fact that this matrix is idempotent has some consequences for tetrahedrally closed line systems in Euclidean spaces. One of the early theorems relating these line systems with near hexagons is known to contain an essential error. We fix this gap here in the special case for geometries having an order. (C) 2020 Elsevier Inc. All rights reserved