Positroid varieties: juggling and geometry

Abstract

While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown-Goodearl-Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen-Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov's and Buch-Kresch-Tamvakis' approaches to quantum Schubert calculus.

Gravity on-shell diagrams

“…There are 2n delta functions in total, four of them give the overall momentum conservation while the remaining 2n − 4 constrain the parameters of the C-matrix. The study of Grassmannians is a vast and active topic in the mathematics community ranging, amongst others, from combinatorics to algebraic geometry [50][51][52][53][54][55]. There is a close connection to on-shell diagrams which was simultaneously discovered both by physicists in the context of scattering amplitudes and by mathematicians (in the math literature JHEP11(2016)136 these diagrams are called plabic graphs) in searching for positive parameterizations of Grassmannians.…”

The expected codimension of a matroid variety

“…where k is the rank of P. A simple computation (which is spelled out in [7]) shows that #I = nk + n, and we know that d I = l(π ) + n, since we are counting the pairs of intervals in the definition of l(π ) and also counting the pairs (I, I ). So, we are left with…”

“…So, for example, the 3 printed in the fourth row and fourth nonempty column indicates that [4,7] = [4, 1] = {4, 5, 6, 1} has rank 3. The underlined entries are the positions of the 1's in the corresponding affine permutation matrix.…”

“…In particular, they are always reduced, irreducible, and Cohen-Macaulay, and unlike general matroid varieties (see Counterexample 2.6) they are always cut out by Plücker variables. Positroids are very well-studied already, and there are several different combinatorial gadgets that have been invented to describe them, some of which are described in [7].…”

“…Counterexample 2.6 Consider the rank-3 matroid A on [7] generated by the conditions that {1, 2, 7}, {3, 4, 7}, and {5, 6, 7} have rank 2. The variety X (A) is not cut out by the ideal ( p 127 , p 347 , p 567 ).…”