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Recognizing Schubert cells. (English) Zbl 1056.14504

From the introduction: Let \(G\) be a simply connected complex semisimple Lie group of rank \(r\) with a fixed Borel subgroup \(B\) and a maximal torus \(H\subset B\). Let \(W=\text{Norm}_G(H)/H\) be the Weyl group of \(G\). The generalized flag manifold \(G/B\) can be decomposed into the disjoint union of Schubert cells \(X^\circ_w=(BwB)/B\), for \(w\in W\). To any weight \(\gamma\) that is \(W\)-conjugate to some fundamental weight of \(G\), one can associate a generalized Plücker coordinate \(p_\gamma\) on \(G/B\). In the case of type \(A_{n-1}\) (i.e., \(G=SL_n)\), the \(p_\gamma\) are the usual Plücker coordinates on the flag manifold. The closure of a Schubert cell \(X^\circ_w\) is the Schubert variety \(X_w\), an irreducible projective subvariety of \(G/B\) that can be described as the set of common zeroes of some collection of generalized Plücker coordinates \(p_\gamma\). It is also known that every Schubert cell \(X^\circ_w\) can be defined by specifying vanishing and/or non-vanishing of some collection of Plücker coordinates. The main two problems studied in this paper are the following.
(1) Describe a given Schubert cell by as small as possible number of equations of the form \(p_\gamma=0\) and inequalities of the form \(p_\gamma\neq 0\).
(2) Suppose a point \(x\in G/B\) is unknown to us, but we have access to an oracle that answers questions of the form: “\(p_\gamma(x)=0\), true or false?” How many such questions are needed to determine the Schubert cell \(x\) is in?
The number of equations of the form \(p_\gamma=0\) needed to define a Schubert variety is generally much larger than its codimension. We show that for a certain Schubert variety \(X_w\) in the flag manifold of type \(A_{n-1}\), one needs exponentially many such equations to define it, even though \(\text{codim}(X_w) \leq\dim (G/B)={n\choose 2}\). Given this kind of “complexity” of Schubert varieties, it may appear surprising that for the types \(A_r,B_r,C_r\), and \(G_2\), we provide a description of an arbitrary Schubert cell \(X^\circ_w\) that only uses \(\text{codim} (X_w)\) equations of the form \(p_\gamma=0\) and at most \(r\) inequalities of the form \(p_\gamma\neq 0\). For the type \(D\), a description of Schubert cells is slightly more complicated. Our main result regarding (2) is an algorithm that recognizes a Schubert cell \(X^\circ_w\) containing an element \(x\). For the types \(A_r,B_r,C_r\), and \(G_2\), our algorithm ends up examining precisely the same Pücker coordinates of \(x\) that appear in the previous result. In the case of type \(A_{n-1}\), recognizing a cell requires testing the vanishing of at most \({n\choose 2}\) Plücker coordinates. Finally, we discuss the problem of presenting a subset of Plücker coordinates whose vanishing pattern determines which cell a point is in.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
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