A semi-recursion for the number of involutions in special orthogonal groups over finite fields

Abstract

Let I ( n ) be the number of involutions in a special orthogonal group SO ( n , F q ) defined over a finite field with q elements, where q is the power of an odd prime. Then the numbers I ( n ) form a semi-recursion, in that for m > 1 we have I ( 2 m + 3 ) = ( q 2 m + 2 + 1 ) I ( 2 m + 1 ) + q 2 m ( q 2 m − 1 ) I ( 2 m − 2 ) . We give a purely combinatorial proof of this result, and we apply it to give a universal bound for the character degree sum for finite classical groups defined over F q .