On the commutant algebras corresponding to the permutation representations of the full collineation groups of PG(k,s) and EG(k,s), s = pr,k ⩾ 2

In this paper, the dimension t and a linear basis of the commutant algebra corresponding to the representation of the full collineation group as matrices permuting the flags (incident point-line or point-hyperplane pairs) have been determined for each one of the four geometries PG(2, s), EG(2, s), PG(k, s), and EG(k, s), s = p^r, k ? 3. For the four geometries, t = 6, 7, 7, and 8, respectively, and the corresponding linear bases are (I, G, B, T, BT, TB), (I, G, B, T, BT, TB, BTB), (I, G, B, T, BT, TB, S), and (I, G, B, T, BT, TB, BTB, S). I, G, B, T are the relationship matrices of James (Ann. Math. Statist.28 (1957), 993–1082) and the matrix S was introduced by Sysoev and Shaikin (Avtomat. i Telemekh.5 (1976), 64–73).