Cyclic symmetry in the Grassmannian

Transcription

1 Cyclic symmetry in the Grassmannian Slides available at www-personal.umich.edu/~snkarp Steven N. Karp, University of Michigan including joint work with Pavel Galashin and Thomas Lam (arxiv: ) September 9th, 07 University of Minnesota, Twin Cities Steven N. Karp (Michigan) Cyclic symmetry in the Grassmannian September 9th, 07 /

2 The Grassmannian Gr The Grassmannian Gr is the set of k-dimensional subspaces V of C n. V := 0 (0,,, ) (, 0,, ) = = [ 0 ] 0 [ ] 0 = = (, ) (, ) (, ) (0, ) 0 (, ) 0 (, 0) (, ) Gr 0,4 {,} =, {,} =, {,4} =, {,} =, {,4} =, {,4} = Given V Gr in the form of a k n matrix, for k-subsets I of {,..., n} let I (V ) be the k k minor of V in columns I. The Plücker coordinates I (V ) are well defined up to a common nonzero scalar. We call V Gr totally nonnegative if I (V ) 0 for all k-subsets I. The set of all such V forms the totally nonnegative Grassmannian Gr 0. We can also view an element V Gr as a vector configuration of n ordered vectors in C k (up to automorphism). Steven N. Karp (Michigan) Cyclic symmetry in the Grassmannian September 9th, 07 / (, 0)

3 The cell decomposition of Gr 0 has a cell decomposition due to Rietsch (998) and Postnikov (007). Each cell is specified by requiring some subset of the Plücker coordinates to be strictly positive, and the rest to equal zero. Gr 0, = 0 Gr 0, (= P 0) = = 0 = 0,, > 0, = 0 = 0, = 0 Postnikov showed that the face poset of Gr 0 is given by circular Bruhat order on decorated permutations with k anti-excedances. Steven N. Karp (Michigan) Cyclic symmetry in the Grassmannian September 9th, 07 / 0

4 Cyclic symmetry of Gr 0 Define the (left) cyclic shift map σ GL n (C) by σ(v) := (v, v,, v n, ( ) k v ) for v = (v,, v n ) C n. Then σ acts on Gr as an automorphism of order n. This cyclic action preserves Gr 0, and gives the cyclic symmetry of the cell decomposition. [ ] [ ] σ 0 0 What are the fixed points of σ? They are the k-dimensional invariant subspaces of C n, spanned by k linearly independent eigenvectors of σ. Theorem (Karp (07+)) There are precisely ( n k) fixed points of σ on Gr, namely, the subspaces spanned by eigenvectors (, ζ, ζ,, ζ n ) for some k distinct nth roots ζ of ( ) k. There is a unique totally nonnegative fixed point V 0, obtained by taking the k eigenvalues ζ closest to on the unit circle. Steven N. Karp (Michigan) Cyclic symmetry in the Grassmannian September 9th, 07 4 /

5 Fixed points of the cyclic shift map Theorem (Karp (07+)) There are precisely ( n k) fixed points of σ on Gr, namely, the subspaces spanned by eigenvectors (, ζ, ζ,, ζ n ) for some k distinct nth roots ζ of ( ) k. There is a unique totally nonnegative fixed point V 0, obtained by taking the k eigenvalues ζ closest to on the unit circle. These values of ζ are e Proof i (k )π n, e Scott (879): I (V 0 ) = i (k )π n r,s I, r<s,, e i (k )π n, e i (k )π n. ( ) s r sin n π for all k-subsets I. Gantmakher, Krein (950): V is in Gr 0 if and only if for all v V, we have Re(v) V and Re(v) changes sign at most k times. Observe that if ζ = e iθ, then Re(, ζ, ζ,, ζ n ) changes sign at times. least n θ π Steven N. Karp (Michigan) Cyclic symmetry in the Grassmannian September 9th, 07 5 /

6 Moment curves and (bi)cyclic polytopes V 0 Gr 0 has an elegant description as a vector configuration. Let θ < θ < < θ n < θ + π be equally spaced on the real line. If k is odd, V 0 is represented by the points θ = θ,, θ n on the curve (, cos(θ), sin(θ), cos(θ), sin(θ),, cos ( k θ), sin ( k θ)) in R k. If we delete the first component we obtain the trigonometric moment curve in R k, which Carathéodory (9) used to define cyclic polytopes. If k is even, we instead use the symmetric moment curve ( ( cos θ), sin ( θ), cos ( θ), sin ( θ),, cos ( k θ), sin ( k θ)) in R k. Barvinok and Novik (007) used this curve to define bicyclic polytopes. e.g. k =, n = 4: θ θ 4 θ θ σ θ 4 θ θ θ +π. These curves also appear as extremal solutions to the isoperimetric problem for convex curves, studied by Schoenberg, Krein, and Nudel man. Steven N. Karp (Michigan) Cyclic symmetry in the Grassmannian September 9th, 07 6 /

7 Quantum cohomology of Gr The cohomology ring H (Gr ) has dimension ( n k). A basis is {[X λ ] : λ k (n k)}, where X λ Gr is the Schubert variety indexed by the partition λ. Multiplication is given by [X λ ] [X µ ] = ν cν λ,ν [X ν], where the cλ,ν ν s are Littlewood Richardson coefficients. We have H (Gr ) = C[e,, e k ]/(h n k+,, h n ), via [X λ ] s λ. The quantum cohomology ring QH (Gr ) is a q-deformation of H (Gr ). By definition, it has the basis {[X λ ] : λ k (n k)}, with multiplication [X λ ] q [X µ ] := d,ν X λ, X µ, X ν d q d [X ν ]. Here X λ, X µ, X ν d counts rational curves passing through generic translates of X λ, X µ, and X ν. Siebert and Tian (994) showed that QH (Gr ) = C[e,, e k, q]/(h n k+,, h n, h n ( ) k q). Steven N. Karp (Michigan) Cyclic symmetry in the Grassmannian September 9th, 07 7 /

8 Quantum cohomology of Gr In unpublished work, Peterson discovered a presentation of the quantum cohomology ring of a general partial flag variety, as the coordinate ring of a Peterson variety. In the case of Gr, this was proved by Rietsch (00). We can identify the Peterson variety P with a certain subvariety of Gr. For q 0, define the q-deformed cyclic shift map σ q GL n (C) by σ q (v) := (v, v,, v n, ( ) k qv ) for v = (v,, v n ) C n. Then σ q acts on Gr as an automorphism of order n. Proposition (Karp (07+)) The Peterson variety P Gr is the union of span{e,, e k } with the sets of fixed points of σ t over all t 0. Explicitly, the isomorphism QH (Gr ) C[P ] sends q to a map P C, whose fiber over t 0 equals the set of fixed points of σ t. Steven N. Karp (Michigan) Cyclic symmetry in the Grassmannian September 9th, 07 8 /

9 Topology of positive spaces Theorem (Galashin, Karp, Lam (07)) Gr 0 is homeomorphic to a closed ball of dimension k(n k). S x y Y = 0 z : x + z =, 0 0 all minors 0 Edelman (98) showed that intervals in S n are shellable. By results of Björner (984), this implies S n is the face poset of a regular CW complex homeomorphic to a ball. Is there a naturally occurring such CW complex? Fomin and Shapiro (000) conjectured that Y n SL n is such a regular CW complex. This was proved by Hersh (04), in general Lie type. Steven N. Karp (Michigan) Cyclic symmetry in the Grassmannian September 9th, 07 9 /

10 Topology of positive spaces Theorem (Galashin, Karp, Lam (07)) Gr 0 is homeomorphic to a closed ball of dimension k(n k). Conjecture (Postnikov (007)) and its cell decomposition form a regular CW complex homeomorphic to a ball. Gr 0 Williams (007): The poset of cells of Gr 0 is shellable. Postnikov, Speyer, Williams (009): Gr 0 is a CW complex. Rietsch, Williams (00): Gr 0 is a regular CW complex up to homotopy (via discrete Morse theory). In order to prove Postnikov s conjecture, one needs to show that the closure of every cell of Gr 0 is homeomorphic to a closed ball. Steven N. Karp (Michigan) Cyclic symmetry in the Grassmannian September 9th, 07 0 /

11 Positive spaces and physics Arkani-Hamed, Bai, Lam (07): a positive geometry is a space equipped with a canonical differential form, which has simple poles (only) at the boundaries of the space. Examples include polytopes and Gr 0 (0, ) y (0, 0) (, 0) x dxdy xy( x y) It appears that the canonical differential forms of several positive geometries have physical significance. So far, three examples are known: amplituhedra scattering amplitudes in planar N = 4 SYM (Arkani-Hamed, Trnka (04)) associahedra scattering amplitudes in bi-adjoint scalar theories (Arkani-Hamed, Bai, He, Yan (07+)); cosmological polytopes wavefunction of the universe in toy models (Arkani-Hamed, Benincasa, Postnikov (07)) Steven N. Karp (Michigan) Cyclic symmetry in the Grassmannian September 9th, 07 /.

12 Showing Gr 0 is homeomorphic to a ball How do we show that a compact, convex subset of R d is homeomorphic to a ball? For Gr 0, we apply the flow given by the vector field σ, which takes V to exp(tσ)(v ) in time t. This flow attracts all of Gr 0 toward V 0. e.g. Gr 0, Steven N. Karp (Michigan) Cyclic symmetry in the Grassmannian September 9th, 07 /

13 Open questions Can we adapt the proof to show that other positive spaces are homeomorphic to balls? What about the closures of cells of Gr 0? In studying mirror symmetry for partial flag varieties, Rietsch (008) introduced the superpotential, a rational function on Gr : n {i+,i+,,i+k,i+k+}. {i+,i+,,i+k} i= It turns out that the critical points of the superpotential are precisely the fixed points of σ. How exactly is the superpotential related to σ? Can we use its gradient flow to show that Gr 0 is homeomorphic to a ball? Thank you! Steven N. Karp (Michigan) Cyclic symmetry in the Grassmannian September 9th, 07 /