 
Summary: NOI~II~
Free Resolutions of Generic Symmetric Matrices
Helmer Aslaksen, EngChye Tan, and Chenbo Zhu*
Department of Mathematics
National University of Singapore
Singapore 0511, Republic of Singapore
Submitted by Shmuel Friedland
ABSTRACT
We give an elementary construction of a finite free resolution of k[P]/In_ t,
where k[ P] is the ring of polynomial functions in the entries of a generic symmetric
n × n matrix P, and In_ 1 is the ideal generated by the n  1 minors of P.
1. INTRODUCTION
Let k be a field of characteristic zero, and let G = O(m, k). Let V = Un,
where U is the standard module of G. Consider the action of G on k[V ], the
space of polynomial functions on V. Denote the algebra of G invariants in
k[V] by k[V] G. We can identify k[V] with the polynomial ring k[X], where
X is an m x n matrix of indeterminates. If P is an n x n symmetric matrix
of indeterminates, it is well known [5, Theorems 2.9.A and 2.17.A] that
k[V ]G ~ k[ XtX ] = k[ e]/Im+ l,
where Ira÷1 C k[P] is the ideal generated by the (m + 1) × (m + 1) minors
