Free resolutions of generic symmetric matrices
For a field \(k\) of characteristic zero let \(G = O(m,k)\) and \(V = U^n\), where \(U\) is the standard module of \(G\). \(G\) acts on \(k[V]\), the space of polynomial functions on \(V\). We can identify \(k[V]\) with the polynomial ring \(k[X]\), where \(X\) is an \(m \times n\) matrix of indeterminates. If \(P\) is an \(n \times n\) symmetric matrix of indeterminates, then \(k[V]^G \simeq k[X^tX] \simeq k[P]/I_{m + 1}\), where \(I_{m + 1} \subset k[P]\) is the ideal generated by the \((m + 1) \times (m + 1)\) minors of \(P\). The authors give an elementary construction of a finite free resolution of \(k[P]/I_{m + 1}\) in the case \(m = n - 2\). The theorem was earlier known [cf. \textit{S. Goto} and \textit{S. Tachibana}, J. Math. Kyoto Univ. 17, 51-54 (1977; Zbl 0375.13004) and \textit{T. Józefiak} [Commentarii Math. Helvet. 53, 595-607 (1978; Zbl 0398.13009)]. However, the present authors' approach is elementary.
scientific article; zbMATH DE number 826706