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Positroid varieties: juggling and geometry. (English) Zbl 1330.14086
What juggling has to do with geometry and how both relate to positroid varieties, the main concern of the impressive paper under review? It is not easy to summarize within the necessarily limited space of a review the content of an article so rich of new ideas, hard techniques and perspectives. Therefore, in this brief report we shall bound ourselves to highlight the most important key words, in the attempt of giving at least a little of the flavour of the article.
Positroid varieties come from positroid. Positroids are combinatorial object, a special kind of matroids. If \([n]\) denotes the interval \(\{i\in{\mathbb N} \,|\, 1 \leq i\leq n\}\), let \[ {\mathcal S}_{k,n}:={[n]\choose k} \] be the set of all the subsets of \([n]\) with \(k\) elements. A matroid \({\mathcal M}\) of rank \(k\) is a subfamily of \({\mathcal S}_{k,n}\), whose elements are called bases, satisfying a certain unique minimum axiom, recalled in Section 3.6 of the paper: for any permutation \(w\in S_n\), there is a unique minimal element of \(w\cdot{\mathcal M}\), in the usual partial order “\(\preceq\)” on \({\mathcal S}_{k,n}\), called minimal base. Let \(\chi\) denote the permutation \((23\ldots n,1)\). Define a sequence \({\mathcal J} ({\mathcal M}) = (J_1,J_2,\ldots,J_n)\) in \({\mathcal S}_{k,n}\), by letting \(J_r\) to be the minimal base of \(\chi^{-r+1}({\mathcal M})\). A matroid of the form \[ \left\{I\in{[n]\choose k}\,|\, \chi^{-r+1}(I)\geq J_r\right\} \] is said to be a positroid.
It turns out that the sequence \({\mathcal J} ({\mathcal M})\) is a \((k,n)\)-sequence of juggling states. In this way, we come to a second key word of this work, which enlightens another aspect of this multi-faceted story. Recall that a \(k\)-plane in \({\mathbb C}^n\) is an equivalence class of \(k\times n\) matrices of rank \(k\), modulo the action of the group \(\mathrm{GL}_k({\mathbb C})\). To each \(k\times n\) matrix, one can associate an infinite array by repeating on the right and on the left the \(n\) columns. Thus, the map \(f:{\mathbb Z}\rightarrow {\mathbb Z}\), defined by \(f(i)=\min\{j\geq i; v_i\in \text{span}(v_i,v_{i+1},\ldots, v_{i+j})\}\) is an affine permutation, i.e. \(f(i+n)=f(i)+n\). The authors explain how to interpret affine permutations in terms of juggling, and this is best explained by their own words: “consider a juggler who is juggling \(k\) balls, one throw every second, doing a pattern of period \(n\). At time \(i\in{\mathbb Z}\), he throws a ball that lands shortly before, to be thrown again at time \(f(i)\). No two balls land at the same time, and there is always a ball available for the next throw. Letting \(t_i = f(i)-i\) to be the throw at time \(i\), the cyclic list of \(n\) numbers \((t_1,\ldots , t_n)\) is called a juggling pattern, or siteswap, a mathematical model developed by many jugglers independently in 1985, of great practical use in the juggling community.”
The point is that the association of an affine permutation to each \(k\times n\) matrix of rank \(k\), depends only on the \(k\)-plane spanned by the rows. Thus, it descends to \(\mathrm{Gr}(k, n)\), the Grasmannians of the \(k\)-planes in \({\mathbb C}^n\), and so provides a complete combinatorial invariant of the strata in the cyclic Bruhat (=Schubert) decomposition. Notice that Schubert varieties can be associated to special kind of positroids: each \(I\in{\mathcal S}_{k,n}\), can be written in the form of a strictly increasing sequence of integers, and so the Schubert variety \(X_I\) associated to the Schubert index \(I\) can be seen associated to the positroid \(\{J : J \geq I\}\).
As well known, Schubert cells with respect to a given complete flag of \({\mathbb C}^n\) form a stratification of the Grassmannian \(\mathrm{Gr}(k,n)\), in the sense that the closure of one open stratum is attained by adding a union of smaller dimensional cells. Open Richardson varieties provide a finer decomposition of \(\mathrm{Gr}(k,n)\), obtained by passing to the common refinement of the Schubert cell decomposition and that obtained by using the opposite Schubert decomposition (i.e., the Schubert decomposition associated to the same flag but reversing the order of the basis of \({\mathbb C}^n\)). The latter is however coarser than the GGMS decomposition, so called after Gel’fand, Goreski, MacPherson and Serganova, [I. M. Gel’fand et al., Adv. Math. 63, 301–316 (1987; Zbl 0622.57014)], which is obtained by intersecting all the Bruhat decompositions relative to all the coordinate flags.
Open positroid varieties in \(\mathrm{Gr}(k,n)\) are intersection of \(n\) Schubert cells, taken with respect to the cyclic rotations of the standard flag. They provide a stratification which stays in between the Richardson and the GGMS one. Positroid varieties are closures of open positroid varieties. The main theorem of the paper, that the authors prove in Section 5, asserts that positroid varieties are projection of the Richardson subvarieties leaving in the full manifold of the flag of \({\mathbb C}^n\). Each positroid variety associated to a grassmannian permutation (i.e. an element of \(S_k\times S_{n-k}\)) arises uniquely in this way. As the authors remark, this result was suspected and reasonable, but the proof reveals to be very and surprsingly difficult.
More properties of positroid varieties are proven in the paper, such as that they are normal, Cohen-Macaulay and have rational singularities. Furthermore, they are very concrete from a geometrical point of view given that their scheme structure is defined by means of vanishing of Plücker coordinates.
The article is organized as follows. Section 1 is the introduction, where the main terminology about stratification of the Grassmannian and juggling are stated. Section 2 reviews some combinatorial preliminaries, more focused on the juggling part in Section 3. Background on Richardson varieties are set in Section 4, while the main results of the paper, regarding positroid varieties are discussed in Section 5. Section 6 is devoted to examples, besides the simplest ones like Schubert varieties, cyclically permuted Schubert varieties and Richardson varieties. The article still goes on with Section 7 and 8, devoted, respectively, to the classical and quantum cohomology of positroid varieties, linking to work of A. Postnikov [Duke Math. J. 128, No. 3, 473–509 (2005; Zbl 1081.14070)] and Buch-Kresch-Tamvakis [A. S. Buch et al., J. Am. Math. Soc. 16, No. 4, 901–915 (2003; Zbl 1063.53090)]. In Section 7, in particular, the authors show that the cohomology class associated to a positroid variety can be represented by the affine Stanley function of its affine permutation, a particular symmetric function introduced by T. Lam [Am. J. Math. 128, No. 6, 1553–1586 (2006; Zbl 1107.05095)]. The paper ends with an impressively rich bibliographical list. To speak the truth, the mathematics described in this paper is not easy – and highly non trivial. However, looking at the article more closely, the patient reader can easily realize that it is mostly self-contained, as it embodies all the notions necessary to follow it.

14M15 Grassmannians, Schubert varieties, flag manifolds
05B35 Combinatorial aspects of matroids and geometric lattices
05E05 Symmetric functions and generalizations
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