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Zbl 0840.13001
Aslaksen, Helmer; Tan, Eng-Chye; Zhu, Chen-bo
Free resolutions of generic symmetric matrices. (English)
[J] Linear Algebra Appl. 230, 21-34 (1995). ISSN 0024-3795

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 $P$.\par 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. {\it S. Goto} and {\it S. Tachibana}, J. Math. Kyoto Univ. 17, 51-54 (1977; Zbl 0375.13004) and {\it T. Józefiak} [Commentarii Math. Helvet. 53, 595-607 (1978; Zbl 0398.13009)]. However, the present authors' approach is elementary.
[W.Wiȩsław (Wrocław)]

MSC 2000:: *13A50 Invariant theory
13F20 Polynomial rings

Keywords: polynomial ring

Citations: Zbl 0375.13004; Zbl 0398.13009

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