The human genome contains hundreds of regions whose patterns of genetic variation indicate recent positive natural selection, yet for most the underlying gene and the advantageous mutation remain unknown. We developed a method, composite of multiple signals (CMS), that combines tests for multiple signals of selection and increases resolution by up to 100-fold. By applying CMS to candidate regions from the International Haplotype Map, we localized population-specific selective signals to 55 kilobases (median), identifying known and novel causal variants. CMS can not just identify individual loci but implicates precise variants selected by evolution.

We construct a type A (1) n−1 geometric crystal on the variety Gr(k, n) × C × , and show that it tropicalizes to the disjoint union of the Kirillov-Reshetikhin crystals corresponding to rectangular tableaux with n − k rows. A key ingredient in our construction is the Z/nZ symmetry on the Grassmannian coming from cyclically shifting the basis of the underlying vector space. We show that a twisted version of this symmetry tropicalizes to combinatorial promotion. Additionally, we use the loop group GLn(C(λ)) to define a unipotent crystal which induces our geometric crystal. We use this unipotent crystal to study the geometric analogues of two symmetries of rectangular tableaux.to the "classical" crystal operators defined on tableaux of all shapes, there is an "affine" crystal operator e 0 defined on rectangular tableaux, corresponding to the action of the additional simple root of the Lie algebra sl n . There are also two crystal-theoretic operations on tensor products of rectangular tableaux-the combinatorial R-matrix and the energy function-which have no analogue in the classical setting ([24]). Ideally, an affine geometric crystal should come with rational lifts of these operations as well.In the one-row case, such an affine geometric crystal has been constructed. The underlying variety is (C × ) n , and a point (z 1 , . . . , z n ) ∈ (C × ) n is the rational analogue of a vector (b 1 , . . . , b n ) ∈ (Z ≥0 ) n which specifies the number of 1 ′ s, 2 ′ s, . . . , n ′ s in a one-row tableau. The affine geometric crystal structure on this variety was described by Kuniba, Okado, Takagi, and Yamada [14], and Yamada [25] found a rational lift of the combinatorial R-matrix for tensor products of one-row tableaux. Lam and Pylyavskyy [16] showed that a certain loop Schur function provides a rational lift of the energy function for tensor products of one-row tableaux.Let Gr(k, n) be the Grassmannian of k-dimensional subspaces in C n , and let B n−k,L denote the Kirillov-Reshetikhin crystal corresponding to rectangular tableaux with n − k rows and L columns. The main contribution of this article is the construction of an affine geometric crystal on the variety Gr(k, n) × C × which tropicalizes to the disjoint union of the crystals B n−k,L .Our starting point is a type A n−1 geometric crystal constructed by Berenstein and Kazhdan [3], which tropicalizes to the "classical" crystal structure on rectangular tableaux with n − k rows. This geometric crystal is defined on a certain subvariety of the group of lower triangular matrices in GL n (see Remark 4.4), and there is a birational isomorphism from this subvariety to Gr(k, n) × C × . The advantage of using the Grassmannian is that it has a natural Z/nZ symmetry coming from cyclically shifting the basis vectors of the underlying n-dimensional vector space. We define the geometric crystal operator e 0 by conjugating the geometric crystal operator e 1 by a twisted version of the cyclic shifting map. We then show (Theorem 5.4) that under a suitable parametrization of the Grassman...

In [Frieden, arXiv:1706.02844], we constructed a geometric crystal on the variety X k := Gr(k, n) × C × which tropicalizes to the affine crystal structure on rectangular tableaux with n − k rows. In this sequel, we define and study the geometric R-matrix, a birational map R :tropicalizes to the combinatorial R-matrix on pairs of rectangular tableaux. We show that R is an isomorphism of geometric crystals, and that it satisfies the Yang-Baxter relation. In the case where both tableaux have one row, we recover a birational action of the symmetric group that has appeared in the literature in a number of contexts. We also define a rational function E : X k 1 × X k 2 → C which tropicalizes to the coenergy function from affine crystal theory.Most of the properties of the geometric R-matrix follow from the fact that it gives the unique solution to a certain equation of matrices in the loop group GLn(C(λ)). Contents 1. Introduction 1 2. Combinatorial background 9 3. Geometric and unipotent crystals 12 4. Positivity and tropicalization 23 5. The geometric R-matrix 28 6. The geometric coenergy function 32 7. One-row tableaux 35 8. Proof of the positivity of the geometric R-matrix 40 9. Proof of the identity g • R = g 45 Appendix A. Planar networks and the Lindström Lemma 50 References 53

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