Affine type A geometric crystal structure on the Grassmannian

International audienceWe construct a type A(1) n−1 affine geometric crystal structure on the Grassmannian Gr(k, n). The tropicalization of this structure recovers the combinatorics of crystal operators on semistandard Young tableaux of rectangular shape (with n − k rows), including the affine crystal operator e 0. In particular, the promotion operation on these tableaux essentially corresponds to cyclically shifting the Plu ̈cker coordinates of the Grassmannian

Geometric Lifting of Affine Type A Crystal Combinatorics

In the first part of this thesis, we construct a type $A_{n-1}^{(1)}$ geometric crystal on the variety $mathbb{X}_k := {rm Gr}(k,n) times mathbb{C}^times$, and show that it tropicalizes to the disjoint union of the Kirillov--Reshetikhin crystals corresponding to rectangular semistandard Young tableaux with $n-k$ rows. A key ingredient in our construction is the $mathbb{Z}/nmathbb{Z}$ symmetry of the Grassmannian which comes from cyclically shifting a basis of the underlying vector space. We show that a twisted version of this symmetry tropicalizes to combinatorial promotion. In the second part, we define and study the geometric $R$-matrix, a birational map $R : mathbb{X}_{k_1} times mathbb{X}_{k_2} rightarrow mathbb{X}_{k_2} times mathbb{X}_{k_1}$ which 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 the birational $R$-matrix of Yamada and Lam--Pylyavskyy. 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 ${rm GL}_n(mathbb{C}(lambda))$.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147545/1/gfrieden_1.pd

Affine type A geometric crystal structure on the Grassmannian

We construct a type A(1) n−1 affine geometric crystal structure on the Grassmannian Gr(k, n). The tropicalization of this structure recovers the combinatorics of crystal operators on semistandard Young tableaux of rectangular shape (with n − k rows), including the affine crystal operator e 0. In particular, the promotion operation on these tableaux essentially corresponds to cyclically shifting the Plu ̈cker coordinates of the Grassmannian

Crystal invariant theory I: Geometric RSK

Berenstein and Kazhdan's theory of geometric crystals gives rise to two commuting families of geometric crystal operators acting on the space of complex $m \times n$ matrices. These are birational actions, which we view as a crystal-theoretic analogue of the usual action of ${\rm SL}_m \times {\rm SL}_n$ on $m \times n$ matrices. We prove that the field of rational invariants (and ring of polynomial invariants) of each family of geometric crystal operators is generated by a set of algebraically independent polynomials, which are generalizations of the elementary symmetric polynomials in $m$ (or $n$) variables. We also give a set of algebraically independent generators for the intersection of these fields, and we explain how these fields are situated inside the larger fields of geometric $R$-matrix invariants, which were studied by Lam and the third-named author under the name loop symmetric functions. The key tool in our proof is the geometric RSK correspondence of Noumi and Yamada, which we show to be an isomorphism of geometric crystals. In an appendix jointly written with Thomas Lam, we prove the fundamental theorem of loop symmetric functions, which says that the polynomial invariants of the geometric $R$-matrix are generated by the loop elementary symmetric functions.Comment: 36 pages. v2: 39 pages, 4 figures. Added part (2) of the main theorem (Theorem 1.1), added Appendix

A Composite of Multiple Signals Distinguishes Causal Variants in Regions of Positive Selection

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.Organismic and Evolutionary BiologyOther Research Uni

Association of IFNGR2 gene polymorphisms with pulmonary tuberculosis among the Vietnamese

Interferon-γ (IFN-γ) is a key molecule of T helper 1 (Th1)-immune response against tuberculosis (TB), and rare genetic defects of IFN-γ receptors cause disseminated mycobacterial infection. The aim of the present study was to investigate whether genetic polymorphisms found in the Th1-immune response genes play a role in TB. In our study, DNA samples were collected from two series of cases including 832 patients with new smear-positive TB and 506 unrelated individuals with no history of TB in the general population of Hanoi, Vietnam. Alleles of eight microsatellite markers located around Th1-immune response-related genes and single nucleotide polymorphisms near the promising microsatellites were genotyped. A set of polymorphisms within the interferon gamma receptor 2 gene (IFNGR2) showed a significant association with protection against TB (P = 0.00054). Resistant alleles tend to be less frequently found in younger age at diagnosis (P = 0.011). Luciferase assays revealed high transcriptional activity of the promoter segment in linkage disequilibrium with resistant alleles. We conclude that the polymorphisms of IFNGR2 may confer resistance to the TB development of newly infected individuals. Contribution of the genetic factors to TB appeared to be different depending on age at diagnosis

Interleukin 12B (IL12B) Genetic Variation and Pulmonary Tuberculosis: A Study of Cohorts from The Gambia, Guinea-Bissau, United States and Argentina

We examined whether polymorphisms in interleukin-12B (IL12B) associate with susceptibility to pulmonary tuberculosis (PTB) in two West African populations (from The Gambia and Guinea-Bissau) and in two independent populations from North and South America. Nine polymorphisms (seven SNPs, one insertion/deletion, one microsatellite) were analyzed in 321 PTB cases and 346 controls from Guinea-Bissau and 280 PTB cases and 286 controls from The Gambia. For replication we studied 281 case and 179 control African-American samples and 221 cases and 144 controls of European ancestry from the US and Argentina. First-stage single locus analyses revealed signals of association at IL12B 3′ UTR SNP rs3212227 (unadjusted allelic p = 0.04; additive genotypic p = 0.05, OR = 0.78, 95% CI [0.61–0.99]) in Guinea-Bissau and rs11574790 (unadjusted allelic p = 0.05; additive genotypic p = 0.05, OR = 0.76, 95% CI [0.58–1.00]) in The Gambia. Association of rs3212227 was then replicated in African-Americans (rs3212227 allelic p = 0.002; additive genotypic p = 0.05, OR = 0.78, 95% CI [0.61–1.00]); most importantly, in the African-American cohort, multiple significant signals of association (seven of the nine polymorphisms tested) were detected throughout the gene. These data suggest that genetic variation in IL12B, a highly relevant candidate gene, is a risk factor for PTB in populations of African ancestry, although further studies will be required to confirm this association and identify the precise mechanism underlying it