On the varieties of pairs of matrices whose product is symmetric.

Abstract

We study the varieties and their coordinate rings of pairs of matrices of indeterminates whose product is symmetric. Hochster and Eagon developed the method of Principal Radical Systems to show that determinantal rings are Cohen-Macaulay normal domain. We use this method to show our rings are Cohen-Macaulay normal domains. We explicitly construct a linear homogeneous system of parameters for our rings and show that the <bold>a</bold>-invariant of our rings is negative. Thus, we reduce the question of F-rationality to that of F-injectivity by a criterion due to Hara and Watanabe. Using Grobner bases techniques and deformation theory, along with a theorem of Conca and Herzog we establish, in several cases, that our rings are F-injective, hence F-rational in these cases. A result of K.E. Smith implies that, in these cases, the corresponding varieties have rational singularities in characteristic 0. Using these results and a criterion due to Kempf, we establish that the varieties of pairs of matrices whose product is symmetric have rational singularities in characteristic 0.