On tangent cones of Schubert varieties (Q1706291)

The authors consider tangent cones of Schubert varieties in the complete flag variety \(\mathcal{F}\), and investigate the problem when the tangent cones of two different Schubert varieties coincide. One can write \(\mathcal{F}=\mathrm{GL}(n,\mathbb{C})/B\) where \(B\) is the Borel subgroup stabilizing the standard flag \(F_0\). The variety \(\mathcal{F}\) is a disjoint union of \(B\)-orbits called Schubert cells. These cells are parametrized by elements of the symmetric group \(S_n\), acting naturally in \(\mathbb{C}^n\), and hence in \(\mathcal{F}\). To each Schubert variety \(X_w\) (namely the closure of a Schubert cell \(C_w=x\cdot F_0\), avec \(w\in S_n\)), one can associate two subsets of the tangent space \(T_{F_0}\mathcal{F}\): \begin{itemize} \item the tangent cone \(T_w\), which is the set of vectors tangent to \(X_w\) at \(F_0\); \item the Zariski tangent space \(Z_w\) which is spanned by \(T_w\). \end{itemize} The tangent cones \(T_w\) are algebraic subvarieties of \(T_{F_0}\mathcal{F}\) which have the same dimensions as \(X_w\). The tangent cone \(T_w\) and tangent space \(Z_w\) coincide if and only if \(X_w\) is not singular point. Given two distinct the Schubert varieties \(X_w\) and \(X_{w'}\), the associated spaces may be equal: \(Z_w = Z_{w'}\) or \(T_w = T_{w'}\) (the second equality implying the first one). [\textit{V. Lakshmibai}, Math. Res. Lett. 2, No. 4, 473--477 (1995; Zbl 0857.14027)] provides an explicit description of the Zariski tangent spaces \(Z_w\) using Chevalley basis. In particular, this results answers the question under which condition two different Schubert varieties have the same Zariski tangent space. On the contrary, the structure of tangent cones \(T_w\), is, by the authors knowledge, not well understood, in particular, the problem of their coincidence is mostly open. The authors study the structure of the tangent cones \(T_w\) with the emphasis on the problem of their coincidence. It was already known that \(T_w = T_{w^{-1}}\) for every permutation \(w\) (this fact was conjectured in [\textit{D. Yu. Eliseev} and \textit{A. N. Panov}, J. Math. Sci., New York 188, No. 5, 596--600 (2013; Zbl 1274.14060); translation from Zap. Nauchn. Semin. POMI 394, 218--225 (2011)]). Moreover all the Schubert varieties corresponding to a Coxeter element of \(S_n\) have the same tangent cone. Note that \(S_n\) has \(2^{n-2}\) Coxeter elements. The authors' main tool is the notion of pillar entries in the rank matrix. Every Schubert cell \(C_w\) is determined by the \((n + 1) \times (n + 1)\) matrix formed by the dimensions \(r_{i, j}\) of the intersections \(V_i \cap V_j^0\) of the subspaces of the flag with the subspace of the standard flag \(F_0\); the corresponding Schubert variety \(W_w\) is determined by inequalities \(\dim(V_i \cap V_j^0 ) \geq r_{i, j}\). The authors prove that the whole rank matrix \((r_{i, j})\) is determined by a relatively small set of entries, which they call pillar entries. Note that the notion of pillar entry is very close (yet different from) Fulton's notion of essential set (see [\textit{W. Fulton}, Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044)]). For example, if \((r_{i, j})\) is the rank matrix corresponding to a permutation \(w\), then the rank matrix corresponding to \(w^{-1}\) is obtained from \((r_{i, j})\) by a transposition. The authors conjecture that if \(T_w = T_{w'}\), then the pillar entries for \(w\) are obtained from pillar entries for \(w\) by a partial transposition. This means that there is a one-to-one correspondence between pillar entries \(r_{i, j}\) and \(r'_{i, j}\) for \(w\) and \(w'\) such that the pillar entry corresponding to \(r_{i, j}\) is either \(r'_{i, j}=r_{i, j}\) or \(r'_{j,i}=r_{i, j}\). However, the converse of this conjecture is false: examples show that a partial transposition of the set of pillar entries may lead to a set of entries which is not the set of pillar entries for any transposition, or is a set of pillar entries coming from a variety of different dimension. Some pillar entries are ``linked'', that is, they can be transposed or not transposed only simultaneously. The authors give a precise definition of a linkage, and hence of ``admissible partial transposition''. Their main result states that an admissible partial transposition of pillars entries of \(w\) provides a set of pillar entries of some \(w'\), and that in this case \(T_w = T_{w'}\). However, examples show that their definition of linkage is not sufficient: there are partial transpositions of pillar entries, which are not admissible in their sense, but which still preserve the tangent cone. The authors present also a formula of (co)dimension of a Schubert variety in terms of the pillar entries of the corresponding rank matrix. Then they present an algorithm that reconstructs a given permutation from the corresponding pillar entries. Finally, the authors, led by the numeric examples, state the following ``2m-conjecture'' which is also closely related with the earlier cited result on Coxeter elements: the number of Schubert varieties with an identical tangent cone is always a power of 2.