Upper triangular matrices and Billiard Arrays

Fix a nonnegative integer d, a field F, and a vector space V over F with dimension d+1. Let T denote an invertible upper triangular matrix in Matd+1(F). Using T we construct three flags on V. We find a necessary and sufficient condition on T for these three flags to be totally opposite. In this case, we use these three totally opposite flags to construct a Billiard Array B on V. It is known that B is determined up to isomorphism by a certain triangular array of scalar parameters called the B-values. We compute these B-values in terms of the entries of T. We describe the set of isomorphism classes of Billiard Arrays in terms of upper triangular matrices.