Relations on Pfaffians: Number of generators (Q1912821)

Let \(X = (x_{ij})_{1 \leq i \leq n, 1 \leq j \leq n}\) denote an alternating matrix of indeterminates. Let \(S = K[X]\) be the polynomial ring in \(X\) over a field \(K\). The ideal \(\text{Pf}_{2 t}\) of \(S\) generated by all \(2t\)-sub Pfaffians of \(X\) is called the generic Pfaffian ideal of order \(2t\). The author describes the number of generators \(\beta_2\) of the first syzygy module of \(\text{Pf}_{2t}\). Let \(\text{char} (K) = p\). He proved that \(\beta_2 = n {n \choose 2t + t} - {n \choose 2t + 2} + \gamma\), where \(\gamma = 0\), provided \(p \neq 2\), resp. \(\gamma = \sum_{i = 1}^{[\log_2 t]} ({n \choose 2^{i + 1} + 2t}\), provided \(p = 2\). Therefore there is no minimal free resolution of generic Pfaffian ideals over \(\mathbb{Z}\) in general. This extends \textit{K. Kurano}'s result [see J. Math. Kyoto Univ. 31, No. 3, 733-742 (1991; Zbl 0781.13006)]. The author's proof depends heavily on another result of Kurano [ibid. 713-731 (1991; Zbl 0781.13005)], saying that \(\beta_{2,j} = 0\) unless \(t + 1 \leq j \leq 2t\), where \(\beta_{2,j}\) denotes the number of generators in degree \(j\). In order to establish a relation between the syzygies of \(\text{Pf}_{2t}\) and the homology of the \(t\)-Schur complex of the identity map there is a generalization of the plethysm formula to complexes. The calculation of the number of generators of the syzygy module turns out by a vanishing theorem on the homology of \(t\)-Schur complexes. Moreover the author describes explicitly the minimal generators of the first syzygy module of \(\text{Pf}_{2t}\).