A new near octagon and the Suzuki tower

We construct and study a new near octagon of order $(2,10)$ which has its full automorphism group isomorphic to the group $\mathrm{G}_2(4){:}2$ and which contains $416$ copies of the Hall-Janko near octagon as full subgeometries. Using this near octagon and its substructures we give geometric constructions of the $\mathrm{G}_2(4)$-graph and the Suzuki graph, both of which are strongly regular graphs contained in the Suzuki tower. As a subgeometry of this octagon we have discovered another new near octagon, whose order is $(2,4)$.Comment: 24 pages, revised version with added remarks and reference

Characterizations of the Suzuki tower near polygons

In recent work, we constructed a new near octagon $\mathcal{G}$ from certain involutions of the finite simple group $G_2(4)$ and showed a correspondence between the Suzuki tower of finite simple groups, $L_3(2) < U_3(3) < J_2 < G_2(4) < Suz$, and the tower of near polygons, $\mathrm{H}(2,1) \subset \mathrm{H}(2)^D \subset \mathsf{HJ} \subset \mathcal{G}$. Here we characterize each of these near polygons (except for the first one) as the unique near polygon of the given order and diameter containing an isometrically embedded copy of the previous near polygon of the tower. In particular, our characterization of the Hall-Janko near octagon $\mathsf{HJ}$ is similar to an earlier characterization due to Cohen and Tits who proved that it is the unique regular near octagon with parameters $(2, 4; 0, 3)$, but instead of regularity we assume existence of an isometrically embedded dual split Cayley hexagon, $\mathrm{H}(2)^D$. We also give a complete classification of near hexagons of order $(2, 2)$ and use it to prove the uniqueness result for $\mathrm{H}(2)^D$.Comment: 20 pages; some revisions based on referee reports; added more references; added remarks 1.4 and 1.5; corrected typos; improved the overall expositio

Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation

Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not neccessarily minimal) permutation representations P. It is unusual but significant to recognize that a P is a Grothendieck's dessin d'enfant D and that most standard graphs and finite geometries G-such as near polygons and their generalizations-are stabilized by a D. In our paper, tripods P -- D -- G of rank larger than two, corresponding to simple groups, are organized into classes, e.g. symplectic, unitary, sporadic, etc (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All the defined geometries G' s have a contextuality parameter close to its maximal value 1.Comment: 19 page

The L₃(4) near octagon

In recent work we constructed two new near octagons, one related to the finite simple group and another one as a sub-near-octagon of the former. In the present paper, we give a direct construction of this sub-near-octagon using a split extension of the group . We derive several geometric properties of this near octagon, and determine its full automorphism group. We also prove that the near octagon is closely related to the second subconstituent of the distance-regular graph on 486 vertices discovered by Soicher (Eur J Combin 14:501-505, 1993)

On the valuations of the near polygon ℍn

We characterize the valuations of the near polygon that are induced by classical valuations of the dual polar space DW (2n - 1, 2) into which it is isometrically embeddable. An application to near 2n-gons that contain as a full subgeometry is given

Loop Optimization of Tensor Network Renormalization: Algorithms and Applications

Loop optimization for tensor network renormalization (loop-TNR) is a real-space renormalization group algorithm suitable for studying 1+1D critical systems. While the original proposal by Yang et al. focused on classical models, we extend this algorithm with new techniques to enable accurate and efficient extraction of conformal data from critical quantum models. Benchmark results are provided for a number of quantum models, including ones described by non-minimal or non-unitary conformal field theories, showcasing both the strengths and limitations of loop-TNR. We discuss the subtle issue of non-analytic finite size effect in quantum lattice models and its impact on loop-TNR, and propose the use of virtual-space transfer-matrix to circumvent it, using the XY model as a demonstration. We then generalize loop-TNR to fermionic systems by incorporating Grassmann numbers, and benchmark the generalized algorithm on the t-V model. Next, we demonstrate a non-trivial application of loop-TNR by studying the 1D domain wall between untwisted and twisted 2D lattice gauge theories of finite groups G. We numerically study such domain walls for G = Z_N (with N<6) using loop-TNR, and discover a large class of gapless models. We also study the physical mechanism for these gapless domain walls and propose quantum field theory descriptions that agree perfectly with our numerical results. By taking advantage of the classification and construction of twisted gauge models using group cohomology theory, we systematically construct general lattice models to realize gapless domain walls for arbitrary finite symmetry group G. Such constructions can be generalized into arbitrary dimensions and might provide us a systematical way to study gapless domain walls and topological quantum phase transitions

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