Europ. J. Combinatorics (1988) 9, 245-247 The Classification of Polarities in Reducible Projective Spaces F. DECLERCK AND F. MAZZOCCA The classification of polarities in irreducible projective spaces is well-known. In this note we classify the polarities in reducible projective spaces. INTRODUCTION A projective space or graphic space [3] is called irreducible if it has no lines incident with exactly two points, i.e. if it has no short lines. If a projective space has short lines it is called a reducible projective space. It is known (see for instance [3]) that a reducible projective space of dimension n is a direct sum P. = P 1 E9 P 2 E9 · · · E9 pm of irreducible projective spaces pi of dimension ni, such that n = L m ni + m- 1. A polarity u in P. is a permutation of the set D(P.) of subspaces of P. such that for any V and W of D(P.), V c W implies W" c V" and V" 2 = V. We will construct a polarity of a reducible projective space. Lett be a fixed integer such that 0 t m. If t -# 0, then we fix in each of the projective spaces pi (1 i t) a polarity ui. If h = m - t -# 0, we suppose that h is even and n,+ 1 = n,+ 2 ; n,+ 3 = nt+4; ... ; nm-l = nm. We denote by r the permutation (t + 1, t + 2) (t + 3, t + 4) ... (m - 1, m) of the set {1, 2, ... , m}. Finally let ui (t < i m) be a correlation between pi and p<(i) such that for a point p of pi and a point q of p<(i) holds (i.e. ui-I = u,(i)). If p is a point of pi we define p" as follows. pi$ . .. $ pi-l $ p"' $ pi+l $ ... $ pm, if i t, p" = pi$ . .. $pi $ p"' $ pi+2 $ ... $ pm, if i = t + k, k odd, { pi$ . .. $ pi-2 $ p"' $pi $ ... $ pm, if i = t + k, k even (k > 0). Then it is obvious that p" is a hyperplane of P •. This map u induces a polarity in P. which we also will denote by u. It is obvious that for any subspace H of P., H" = npeH p" and that dimH" = n - 1 - dimH. Any polarity u defined as above will be called a (t, r)-polarity of P •. We will prove that there are no other polarities. THEOREM 1. Let u be a polarity of Pn. If Hi = (PI E9 ... E9 pi-I E9 pi+I E9 ... E9 pm)" Pj, with i -# j, then Hi = Pj. 245 0195-6698/88/030245 +03 $02.00/0 © 1988 Academic Press Limited