In this article we explain how the coordinate ring of each (open) Schubert variety in the Grassmannian can be identified with a cluster algebra, whose combinatorial structure is encoded using (target labelings of) Postnikov's plabic graphs. This result generalizes a theorem of Scott [Proc. Lond. Math. Soc. (3) 92 (2006) 345-380] for the Grassmannian, and proves a folklore conjecture for Schubert varieties that has been believed by experts since Scott's result [Proc. Lond. Math. Soc. (3) 92 (2006) 345-380], though the statement was not formally written down until Muller-Speyer explicitly conjectured it [Proc. Lond. Math. Soc. (3) 115 (2017) 1014-1071]. To prove this conjecture we use a result of Leclerc [Adv. Math. 300 (2016) 190-228] who used the module category of the preprojective algebra to prove that coordinate rings of many Richardson varieties in the complete flag variety can be identified with cluster algebras. Our proof also uses a construction of Karpman [J. Combin. Theory Ser. A 142 (2016) 113-146] to build plabic graphs associated to reduced expressions. We additionally generalize our result to the setting of skew-Schubert varieties; the latter result uses generalized plabic graphs, that is, plabic graphs whose boundary vertices need not be labeled in cyclic order. Contents

The hypersimplex ∆ k+1,n is the image of the positive Grassmannian Gr ≥0 k+1,n under the moment map. It is a polytope of dimension n − 1 which lies in R n . Meanwhile, the amplituhedron A n,k,2 (Z) is the projection of the positive Grassmannian Gr ≥0 k,n into the Grassmannian Gr k,k+2 under the amplituhedron map Z. It is not a polytope, and has full dimension 2k inside Gr k,k+2 . Nevertheless, as was first discovered in [LPW20], these two objects appear to be closely related via T-duality. In this paper we use the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes -images of positroid cells of Gr ≥0 k+1,n under the moment map -translate into sign conditions characterizing the T-dual Grasstopes -images of positroid cells of Gr ≥0 k,n under the amplituhedron map. Moreover, we subdivide the amplituhedron into chambers enumerated by the Eulerian numbers, just as the hypersimplex can be subdivided into simplices enumerated by the Eulerian numbers. We use this property to prove one direction of the conjecture of [LPW20]: whenever a collection of positroid polytopes gives a triangulation of the hypersimplex, the T-dual Grasstopes give a triangulation of the amplituhedron. Along the way, we prove several more conjectures: Arkani-Hamed-Thomas-Trnka's conjecture that A n,k,2 (Z) can be characterized using sign conditions, and Lukowski-Parisi-Spradlin-Volovich's conjectures about characterizing generalized triangles (the 2k-dimensional positroid cells which map injectively into the amplituhedron A n,k,2 (Z)), and m = 2 cluster adjacency. Finally, we discuss new cluster structures in the amplituhedron.

Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan-Lusztig immanants, which are indexed by permutations, involve q = 1 specializations of Type A Kazhdan-Lusztig polynomials, and were defined by Rhoades and Skandera in [Rhoades and Skandera, 2006]. Using results of [Haiman, 1993] and [Stembridge, 1991], Rhoades and Skandera showed that Kazhdan-Lusztig immanants are nonnegative on matrices whose minors are nonnegative. We investigate which Kazhdan-Lusztig immanants are positive on k-positive matrices (matrices whose minors of size k × k and smaller are positive). The Kazhdan-Lusztig immanant indexed by v is positive on k-positive matrices if v avoids 1324 and 2143 and for all non-Our main tool is Lewis Carroll's identity.

The Grassmannian is a disjoint union of open positroid varieties Π • µ , certain smooth irreducible subvarieties whose definition is motivated by total positivity. The coordinate ring C[Π •µ ] is a cluster algebra, and each reduced plabic graph G for Π • µ determines a cluster. We study the effect of relabeling the boundary vertices of G by a permutation ρ. Under suitable hypotheses on the permutation, we show that the relabeled graph G ρ determines a cluster for a different open positroid variety Π • π . As a key step in the proof, we show that Π • π and Π • µ are isomorphic by a nontrivial twist isomorphism. Our constructions yield a family of cluster structures on each open positroid variety, given by plabic graphs with appropriately permuted boundary labels. We conjecture that the seeds in all of these cluster structures are related by a combination of mutations and Laurent monomial transformations involving frozen variables, and establish this conjecture for (open) Schubert and opposite Schubert varieties. As an application, we also show that for certain reduced plabic graphs G, the "source" cluster and the "target" cluster are related by mutation and Laurent monomial rescalings.

Suppose that T is an acyclic r-uniform hypergraph, with r ≥ 2. We define the (t-color) chromatic Ramsey number χ(T, t) as the smallest m with the following property: if the edges of any m-chromatic r-uniform hypergraph are colored with t colors in any manner, there is a monochromatic copy of T . We observe that χ(T, t) is well defined andis the t-color Ramsey number of H. We give linear upper bounds for χ(T, t) when T is a matching or star, proving that for r ≥ 2, k ≥ 1, t ≥ 1, χ(M r k , t) ≤ (t − 1)(k − 1) + 2k and χ(S r k , t) ≤ t(k − 1) + 2 where M r k and S r k are, respectively, the r-uniform matching and star with k edges.The general bounds are improved for 3-uniform hypergraphs. We prove that χ(M 3 k , 2) = 2k, extending a special case of Alon-Frankl-Lovász' theorem. We also prove that χ(S 3 2 , t) ≤ t + 1, which is sharp for t = 2, 3. This is a corollary of a more general result. We define H [1] as the 1-intersection graph of H, whose vertices represent hyperedges and whose edges represent intersections of hyperedges in exactly one vertex. We prove that χ(H) ≤ χ(H [1] ) for any 3-uniform hypergraph H (assuming χ (H [1] ) ≥ 2). The proof uses the list coloring version of Brooks' theorem.