Asymptotics of the number of involutions in finite classical groups

Abstract

Abstract Answering a question of Geoff Robinson, we compute the large n limiting proportion of i GL ⁢ ( n , q ) / q ⌊ n 2 / 2 ⌋ {i_{\mathrm{GL}}(n,q)/q^{\lfloor n^{2}/2\rfloor}} , where i GL ⁢ ( n , q ) {i_{\mathrm{GL}}(n,q)} denotes the number of involutions in the group GL ⁢ ( n , q ) {\mathrm{GL}(n,q)} . We give similar results for the finite unitary, symplectic, and orthogonal groups, in both odd and even characteristic. At the heart of this work are certain new “sum = product” identities. Our self-contained treatment of the enumeration of involutions in even characteristic symplectic and orthogonal groups may also be of interest.