Representation of A Lie Group - Formulaic Examples

Formulaic Examples

Let F_q be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the F_q-rational points of a connected reductive group G defined over F_q. For example, if n is a positive integer GL(n, F_q) and SL(n, F_q) are finite groups of Lie type. Let, where I_n is the n×n identity matrix. Let

Then Sp(2,F_q) is a symplectic group of rank n and is a finite group of Lie type. For G = GL(n, F_q) or SL(n, F_q) (and some other examples), the standard Borel subgroup B of G is the subgroup of G consisting of the upper triangular elements in G. A standard parabolic subgroup of G is a subgroup of G which contains the standard Borel subgroup B. If P is a standard parabolic subgroup of GL(n, F_q), then there exists a partition (n₁, …, n_r) of n (a set of positive integers such that ) such that, where has the form

and

where denotes arbitrary entries in .