Minuscule Schubert varieties of exceptional type (Q2682806)

This article is discussing the study of exceptional minuscule Schubert varieties from the perspective of the theory of minimal free resolutions. The authors obtain the defining ideals of the intersections of these Schubert varieties with the big open cell, as well as their resolutions. The main findings of the study are summarized in the following Theorem. {Theorem} (Theorem 1.1 in the article). Let \(G\) be a reductive group of exceptional type. Let \(P \subset G\) a standard parabolic sub-group stabilizing a minuscule fundamental weight \(\omega_i\). Denote by \(V (\omega_i)\) the fundamental \(G\)-representation over \(\mathbb C\) of highest weight \(\omega_i\), and let \(v_{\omega_i} \in V (\omega_i)\) be a highest weight vector. Let \(B \subset P\) be the Borel subgroup containing \(P\) and let \(B^-\) be the opposite Borel subgroup. In particular, the intersection \(B\cap B^-\) coincides with the maximal torus \(T\). Let \(U = B^- \cdot v_{\omega_i}\) be the opposite big open cell \(U \subset \mathbb P(V(\omega_i))\). Then for any given minuscule Schubert variety \(X_\sigma \subset G/P\), the intersection \(Y_\alpha = X_\alpha\cap U\) is as described in Sections 3 and 4. In particular, for \(\sigma\) of type \(E_6\) or as described in Section 4, \(Y_\sigma\) is a complete intersection in one of the following: \begin{itemize} \item[a.] \(Y_\sigma\) is a complete intersection -- the minimal free resolution of its coordinate ring is a Koszul complex. \item[b.] in the codimension three variety of submaximal Pfaffians of a skew symmetric matrix. \item[c.] in the variety of pure spinors in \(V(\omega_4,D_5)\). \item[d.] in a variety of complexes. \item[e.] in a Huneke-Ulrich Gorenstein ideal of codimension 5 and deviation 2. \item[f.] in the variety defined by the vanishing of the \(2\times 2\) minors of a \(2\times 3\) generic matrix. \item[g.] in the variety defined by the vanishing of \(4\times 4\) Pfaffians of a \(6\times 6\) skew-symmetric matrix. \end{itemize} The paper is organized in three sections, with the first section dedicated to preliminaries. The second and third sections are dedicated to \(E_6\) and \(E_7\), respectively, and they determine the defining equations for all Schubert varieties intersected with the big open cell, and compute their Hilbert polynomials. The authors manipulate the given equations by restricting the fundamental representation corresponding to P to the Levi subgroup of G of type D5 to calculate the minimal free resolutions of their defining ideals and Hilbert functions. The authors also obtain generators of an ideal whose associated resolution is precisely \(R_\alpha\).