Positroids and Totally Nonnegative Complex Grassmanian

Question

Recently I begin working on matroids, in particular to a generalization of oriented matroids to the complex case. I found on arxiv the following interesting articles:

1)Alexander Postnikov: Total positivity, Grassmanians and networks

2)Federico Ardila, Felipe Rincón, Lauren Williams: Positively oriented matroids are realizable

3)Federico Ardila, Felipe Rincón, Lauren Williams: Positroids and non-crossing partitions http://arxiv.org/abs/1308.2698

I recall that a positroid of rank $k$ on $n$ elements is a matroid $M$ (of rank $k$ on $n$ elements) if there exist a real totally nonnegative $k\times n$ matrix which realize $M.$

In particular in (3) it is shown in detail a one-to-one correspondence between positroids of rank $k$ over $n$ elements and the cell decomposition of the totally nonnegative grassmanian $\mathcal{G}_{\mathbb{R}}^{\geq}(k,n).$ I hope not to make mistake with notations.

By the way, I'm interested in a generalization to the complex case. However, I have some problem in pointing out work, in particular I don't well understand what is the counterpart of the totally nonnegative Grassmanian in the complex case.

Answer 1

This answer basically repeats what Sam Hopkins said in comments: Allen Knutson, Thomas Lam and I address this question in Positroid Varieties: Juggling and Geometry. We build a stratification of the complex Grassmannian $G(k,n)$ indexed by positroids which has many different descriptions. We write $\Pi^{\circ}(M)$ for the locally closed statum corresponding to the positroid $M$ and $\Pi(M)$ for its closure. This is a stratification, meaning that $G(k,n)$ is the dijoint union of the $\Pi^{\circ}(M)$ and that $\Pi(M) = \bigcup_{M' \leq M} \Pi^{\circ}(M)$ where $M'$ is the closure relation from Postnikov's paper.

The strata can be described in various ways:

• The $\Pi^{\circ}(M)$ are the nonempty intersections of circularly permuted Schubert cells, and $\Pi(M)$ are the intersections of the corresponding circularly permuted Schubert varieties.

• The $\Pi(M)$ are the complete list of Frobenius split subvarieties for the standard Frobenius splitting on the Grassmannian. This splitting is the unique splitting for which the $n$ rotations of standard Schubert divisor are split. (Note that this is an anti-canonical divisor: The set of split hypersurfaces is always of the form anti-canonical minus effective, so this is in some sense a maximal splitting.)

• $\Pi(M)$ is the Zariski closure of the locus in $G(k,n)_+$ which Postnikov associates to $M$. Going in the other direction, $\Pi(M) \cap G(k,n)_+$ and $\Pi^{\circ}(M) \cap G(k,n)_+$ are the closed and locally closed strata in $G(k,n)_+$ which Postnikov associates to $M$.

• Representing a point in $G(k,n)$ by a $k \times n$ matrix $X$, write $r_{ab}(X)$ for the rank of the submatrix with columns $\{ a,a+1, \ldots, b \}$, with the indices wrapping around modulo $n$. This is a well defined function on $G(k,n)$. Write $r_{ab}(M)$ for the rank function of the positroid $M$ evaluated on $\{ a,a+1, \ldots, b \}$. Then $\Pi^{\circ}(M) = \{ X : r_{ab}(X) = r_{ab}(M) \}$ and $\Pi(M) = \{ X : r_{ab}(X) \leq r_{ab}(M) \}$.

• The saturated ideal of $\Pi(M)$, in the homogenous coordinate ring of $G(k,n)$, is generated by the Plucker coordinates $\Delta_I$ for $I$ a nonbasis of $M$.

• For any Richardson variety $R_u^w$ in $\mathcal{F} \ell(n)$, the projection $\pi(R_u^w)$ in $G(k,n)$ is of the form $\Pi(M)$ where $\pi$ is the standard projection $\mathcal{F} \ell(n) \to G(k,n)$ and there is a combinatorial rule for converting $(u,w)$ to $M$ (see our paper). For any $M$, we can choose $(u,w)$ so that the open Richardson $\mathring{R}_u^w$ maps isomorphically to $\Pi^{\circ}(M)$ and the closed Richardson $R_u^w$ maps birationally to $\Pi(M)$ with $\pi^{\ast} \mathcal{O}_{R_u^w} = \mathcal{O}_{\Pi(M)}$ and $R^i \pi^{\ast} \mathcal{O}_{R_u^w} = 0$ for $i>0$. The pairs $(u,w)$ for which this occurs are those for which $u \leq_k w$ in the sense of Bergeron and Sotille. This is analogous to a result about $G(k,n)_+$ and $\mathcal{F} \ell(n)_+$ due to Konnie Rietsch and George Lusztig.

Basically, every description you know for the positive Grassmannian has a complex analogue EXCEPT $\Pi^{\circ}(M)$ being the locus of points whose matroid is $M$. It is larger than that locus.