A cautionary tale of 6 polynomials in 10 variables _______________________________________ Some relevant, useful, or just pretty facts concerning Sp(2n) --- (0) Given n, all nonsingular real skew-symmetric 2n x 2n matrices are equivalent; so fix on the convenient canonical choice K == [[O_n, -I_n], [I_n, O_n]], where O_n, I_n denote n x n zero, unit matrices resp. (1) The "symplectic group" Sp(2n) comprises real 2n x 2n matrices M such that M' K M = K, where M' denotes transpose of M. (2) For M in Sp(2n), the determinant |M| = +1 always, -1 never; (3) Sp(2n) is generated by products of at most 4n "transvections" I + t U' U K, where U ranges over real nonzero 2n-vectors |R^(2n), t over reals |R. (4) The dimension of Sp(2n) equals (2n+1)_C_2 = (2n+1)n; (5) If M = Q S denote the polar decomposition of symplectic M into a unique product of orthogonal Q matrix and symmetric matrix S, then Q,S must also be symplectic; (6) Sp(2n) is a connected Lie group, though not compact; (7) PSp(2n), its "projective symplectic" quotient with scalars factored out, is simple; (8) Mp(2n), its "metaplectic" double cover, is simply connected. (9) Mp(2n) has no (faithful, finite order) matrix representation. I became diverted from an earlier investigation by the unexpected and rather pretty result (6). This familiar situation was doubtless exacerbated by (6) being just about the only item in section 1 of Eckhard Meinrenken's "Symplectic Geometry" (free download online) that I could grasp at sight, and by its proof appearing simple and elegant but quite incomprehensible --- what do diffeomorphisms have to do with a theorem in elementary matrix algebra? So I sat down to investigate (6) by myself. And the first question that occurred to me was: what are the dimensions of the orthogonal and symmetric factors in the polar decomposition of a symplectic matrix? Now physicists by and large don't seem to be terribly interested in Lie groups as such: they are often perfectly content with the Lie algebra --- the vector-space tangent to the identity transformation --- the dimension of which must equal that of the group manifold. Standard results in the literature assert that (10) Lie algebra of Sp(2n) comprises T with T' K + K T = 0; so T = [[A, B], [C, -A']] with B,C symmetric, dimension = 2 (n+1)_C_2 + n^2 = (2n+1)_C_2 = 2 n^2 + n; and (11) Lie algebra of O(2n) comprises T with T' I + I T = 0; so T skew-symmetric; dimension = (2n)_C_2 = 2 n^2 - n; hence (12) Lie algebra of orthogonal symplectic SpO(2n) comprises T = [[A, -B][B, A]], B symm, A skew-symm; dimension = (n+1)_C_2 + n_C_2 = n^2; and (13) By taking the complement of vector space (12) in (11), symmetric symplectic matrices have dimension = 2 n^2 + n - n^2 = n^2 + n = (n+1)_C_2. Much as I mistrust all this high-falutin' abstraction, I'm beginning to wonder if they have a point --- though it should be said in my defence that symmetric matrices, not being a group, are less than amenable to the method. So, just as as a check, I pressed on to try to verify (13) directly. Studiously avoiding an earlier misfortune [involving much earnest investigation of the case n = 1, prior to eventual realisation that actually Sp(2) = SL(2)], I started off with n = 2, for which (13) predicts dimension 6. Setting M = [[z,a,b,d], [a,y,c,e], [b,c,x,f], [d,e,f,w]] and evaluating M' K M = K yields 6 constraints, all of which must vanish: P_1 = b^2 + c*d - f*a - x*z + 1, P_2 = e^2 + c*d - f*a - w*y + 1, P_3 = b*c + e*c - a*x - f*y, P_4 = b*d + e*d - a*w - f*z, P_5 = b*f - e*f + c*w - d*x, P_6 = b*a - e*a - c*z + d*y. It's easy to see informally that these are functionally independent: for each P1,...P6 in turn, set to zero all but b; e; f,y; f,z; d,x; d,y respectively, to yield an expression varying while the other 5 vanish. Or more formally, verify that the 6 x 10 differential matrix [d P_j / d V_i] of linear polynomials in the variables V = [a,b,c,d,e,f,z,y,x,w] has full rank 6. Hence the manifold should have dimension 10-6 = 4; and more generally, dimension = (2n+1)_C_2 - 4 n_C_2 = 2n (?) Comparison with (13) now reveals to the astute observer a trifling discrepancy --- a shortfall of 6-4 = 2 in my bull-at-the-gate estimate for the symmetric symplectic dimension when n = 2; and more generally, of (n+1)n - 2n = (n-1)n = n_C_2. No prizes for guessing which argument is shambolically addled. But just exactly at what point did its misguided author's powers of ratiocination desert him this time round? Fred Lunnon