Homogeneous tuples of operators and holomorphic discrete series representation of some classical groups

Let T = (T1....., Tn) be a n-tuple of bounded linear operators on a fixed Hilbert space . H and let φ be a biholomorphic automorphism of Ω, the joint spectrum of T. In this paper, we consider those n-tuples T for which the joint spectrum Ω is of the form G/K, a bounded symmetric domain. Let φ be any biholornorphic automorphism of the domain Ω. Define, phi(T) via a suit­ able functional calculus and call a n-tuple of operators T homogeneous if φ(T) .is simultaneously unitarily equivalent to T for every automorphism φ of Ω. For each homogeneous operator T, let Uφ be a unitary] operator implimenting this equivalence. We obtain a characterisation of all the homogeneous operators Cowen-Douglas class and show that it is possible to choose the unitary Uφ in such a way that the map φ→ Uφ -1 is a unitary representation of the group of of biholomorpic automorphisms of Ω

On the order of a non-abelian representation group of a slim dense near hexagon

We show that, if the representation group $R$ of a slim dense near hexagon $S$ is non-abelian, then $R$ is of exponent 4 and $|R|=2^{\beta}$, $1+NPdim(S)\leq \beta\leq 1+dimV(S)$, where $NPdim(S)$ is the near polygon embedding dimension of $S$ and $dimV(S)$ is the dimension of the universal representation module $V(S)$ of $S$. Further, if $\beta =1+NPdim(S)$, then $R$ is an extraspecial 2-group (Theorem 1.6)

Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order

Let Q0 be the classical generalized quadrangle of order q = 2n (n ≥ 2) arising from a non-degenerate quadratic form in a 5-dimensional vector space defined over a finite field of order q. We consider the rank two geometry X having as points all the elliptic ovoids of Q0 and as lines the maximal pencils of elliptic ovoids of Q0 pairwise tangent at the same point. We first prove that X is isomorphic to a 2-fold quotient of the affine generalized quadrangle Q Q0, where Q is the classical (q, q2)- generalized quadrangle admitting Q0 as a hyperplane. Further, we classify the cliques in the collinearity graph is either a line of X or it consists of 6 or 4 points of X not contained in any line of X, accordingly as n is odd or even.We count the number of cliques of each type and show that those cliques which are not contained in lines of X arise as subgeometries of Q defined over F2

Codes associated with generalized polygons

This article does not have an abstract

Two characterizations of even order miquelian inversive planes

Using coding theory, we prove theorems 1 and 2 stated below