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I was reading Postnikov's paper [TOTAL POSITIVITY, GRASSMANNiANS, AND NETWORKS][1] when I came across the definition of the totally nonnegative Grassmannian $Gr_{kn}^{tnn} \subset Gr_{kn}$ as the quotient $Gr_{kn}^{tnn} = GL_k^{+}\backslash Mat^{tnn}_{kn}$,

where $Mat^{tnn}_{kn}$ is the set of real $k\times n$-matrices $A$ of rank $k$ with nonnegative maximal minors $\Delta_I(A)\geq 0$ and $GL_k^+$ is the group of $k\times k$-matrices with positive determinant.

My question is, what is the meanig of the quotient here? And how do we visualize this quotient geometrically? I understand that the grassmannian $Gr_{kn}$ is the space of k-subspaces of n-dimensional linear space $V$.

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1 Answer 1

up vote 9 down vote accepted

There are two issues here:

  1. You can represent a $k$-subspace of $V$ by a $k \times n$ matrix whose rows are a basis for the subspace, but this way of representing the subspace is not unique, since your subspace has many different bases. The usual way to handle this is to consider cosets of $k\times n$ matrices under the left action of $GL_k$, since two matrices will be in the same coset iff one is obtained from the other by row operations that preserve the span of the rows. The notation $GL_k \setminus Mat_{kn}$ (where I've left off the + from $GL_k$ and the tnn from $Mat_{kn}$) means the left cosets of the $k\times n$ matrices under the action of $GL_k$.

  2. Postnikov is restricting to those cosets which include a $k\times n$ matrix that is "totally nonnegative", which means that all of its minors are nonnegative. (Warning: people often study totally positive matrices where all minors must actually be positive, but Postnikov, Lusztig and others have been considering real matrices where minors can be either 0 or positive.) The "tnn" is notation for this. And in this case he takes $GL_k^+$ cosets instead of $GL_k$ cosets, since he only needs to identify totally nonnegative $k\times n$ matrices with each other.

Edit: I forgot to say anything about how to visualize this space. People believe it is a ball, but have not so far proven this (to my knowledge). It is the image of a map from a polytope Postnikov introduced in the paper you are reading. The restriction of this map to the interior of the polytope is a homeomorphism, but many points on the boundary of the polytope get identified with each other. My impression from talking with people who've worked with this space is that they can visualize it to varying degrees, but mainly I think try to understand it by trying to understand this map really well. Another paper you might find helpful is:

MR2525057 Reviewed Postnikov, Alexander; Speyer, David; Williams, Lauren Matching polytopes, toric geometry, and the totally non-negative Grassmannian. J. Algebraic Combin. 30 (2009), no. 2, 173–191. (Reviewer: T. Oda) 20G20 (05B35 13F60 14M25 52B70)

This proves it has a CW decomposition and relates this space you are considering to the totally nonnegative part of a toric variety. Perhaps others here will have more to say about how to visualize this space.

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One other comment: I've done some work on a related (but different) space than the one you are studying, where I deal with actual totally nonnegative matrices rather than with cosets. –  Patricia Hersh Oct 20 '12 at 14:10
    
@Patricia: Thanks a lot! Your explanation on the group action is really helpful! –  Harry Huang Oct 20 '12 at 15:23
    
Glad to hear that! –  Patricia Hersh Oct 21 '12 at 2:05

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