Kac's question, planar isospectral pairs and involutions in projective space: II. Classification of generalized projective isospectral data

In Am. Math. Monthly ( 73 1-23 (1966)), Kac asked his famous question 'Can one hear the shape of a drum?', which was eventually answered negatively in Gordon et al (1992 Invent. Math. 110 1-22) by construction of planar isospectral pairs. Giraud (2005 J. Phys. A: Math. Gen. 38 L477 - 83) observed that most of the known examples can be generated from solutions of a certain equation which involves a set of involutions of an n-dimensional projective space over some finite field. He then generated all possible solutions for n=2, when the involutions fix the same number of points. In Thas (2006 J. Phys. A: Math. Gen. 39 L385-8) we showed that no other examples arise for any other dimension, still assuming that the involutions fix the same number of points. In this paper we study the problem for involutions not necessarily fixing the same number of points, and solve the problem completely.