Configurations of lines and models of Lie algebras

The automorphism groups of the 27 lines on the smooth cubic surface or the 28 bitangents to the general quartic plane curve are well-known to be closely related to the Weyl groups of $E\_6$ and $E\_7$. We show how classical subconfigurations of lines, such as double-sixes, triple systems or Steiner sets, are easily constructed from certain models of the exceptional Lie algebras. For ${\mathfrak e}\_7$ and ${\mathfrak e}\_8$ we are lead to beautiful models graded over the octonions, which display these algebras as plane projective geometries of subalgebras. We also interpret the group of the bitangents as a group of transformations of the triangles in the Fano plane, and show how this allows to realize the isomorphism $PSL(3,F\_2)\simeq PSL(2,F\_7)$ in terms of harmonic cubes.Comment: 31 page

On the Coble quartic

We review and extend the known constructions relating Kummer threefolds, Gopel systems, theta constants and their derivatives, and the GIT quotient for 7 points in P^2 to obtain an explicit expression for the Coble quartic. The Coble quartic was recently determined completely by Q.Ran, S.Sam, G.Schrader, and B.Sturmfels, who computed it completely explicitly, as a polynomial with 372060 monomials of bidegree (28,4) in theta constants of the second order and theta functions of the second order, respectively. Our expression is in terms of products of theta constants with characteristics corresponding to Gopel systems, and is a polynomial with 134 terms. Our approach is based on the beautiful geometry studied by Coble and further investigated by Dolgachev and Ortland, and highlights the geometry and combinatorics of syzygetic octets of characteristics, and the corresponding representations of the symplectic group. One new ingredient is the relationship of Gopel systems and Jacobian determinants of theta functions. In genus 2, we similarly obtain a short explicit equation for the universal Kummer surface, and relate modular forms of level two to binary invariants of six points on P^1

On the Coble quartic

We review and extend the known constructions relating Kummer threefolds, G¨opel systems, theta constants and their derivatives, and the GIT quotient for 7 points in P^2 to obtain an explicit expression for the Coble quartic. The Coble quartic was recently determined completely in [RSSS12], where it was computed completely explicitly, as a polynomial with 372060 monomials of bidegree (28, 4) in theta constants of the second order and theta functions of the second order, respectively. Our expression is in terms of products of theta constants with characteristics corresponding to G¨opel systems, and is a polynomial with 134 terms. Our approach is based on the beautiful geometry studied by Coble and further investigated by Dolgachev and Ortland and highlights the geometry and combinatorics of syzygetic octets of characteristics, and the corresponding representations of Sp(6, F_2). One new ingredient is the relationship of G¨opel systems and Jacobian determinants of theta functions. In genus 2, we similarly obtain a short explicit equation for the universal Kummer surface, and relate modular forms of level two to binary invariants of six points on P^

Operation of weaving partial Steiner triple systems

We introduce an operation of a kind of product which associates with a partial Steiner triple system another partial Steiner triple system, the starting one being a quotient of the result. We discuss relations of our product to some other product-like constructions and prove some preservation/non-preservation theorems. In particular, we show series of anti-Pasch Steiner triple systems which are obtained that way

On the Pauli graphs of N-qudits

A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of N-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding Pauli graph are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin's square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) N-qubit Pauli graph is shown to be pseudo-geometric, i. e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle Q(4, 3), the dual ofW(3).Comment: 17 pages. Expanded section on two-qutrits, Quantum Information and Computation (2007) accept\'