Flag Actions and Representations of the Symplectic Group

A flag of a finite dimensional vector space V is a nested sequence of subspaces of V. The symplectic group of V acts on the set of flags of V. We classify the orbits of this action by defining the incidence matrix of a flag of V and showing that two flags are in the same orbit precisely when they have the same incidence matrix. We give a formula for the number of orbits of a certain type and discuss how to list the incidence matrices of all orbits. In the case in which V is a vector space over a finite field, we discuss the permutation representations of the symplectic group of V corresponding to these orbits. For the case in which V = F4q , we compute the conjugacy classes of the symplectic group of V and the values of the characters of the previously discussed permutation representations.