scholarly journals Frobenius splitting of Schubert varieties of semi-infinite flag manifolds

2021 ◽  
Vol 9 ◽  
Author(s):  
Syu Kato
Keyword(s):  
Positive Characteristic ◽  
Formal Model ◽  
Flag Manifold ◽  
Schubert Varieties ◽  
Flag Manifolds ◽  
Line Bundles ◽  
Closed Field ◽  
Flag Varieties ◽  
Fixed Degree ◽  
Frobenius Splitting

Abstract We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field ${\mathbb K}$ of characteristic $\neq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind-)scheme structure, its projective coordinate ring has a $\mathbb {Z}$ -model and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. 371 no.2 (2018)]) when $\mathsf {char}\, {\mathbb K} =0$ or $\gg 0$ , and the higher-cohomology vanishing of their nef line bundles in arbitrary characteristic $\neq 2$ . Some particular cases of these results play crucial roles in our proof [47] of a conjecture by Lam, Li, Mihalcea and Shimozono [60] that describes an isomorphism between affine and quantum K-groups of a flag manifold.

scholarly journals Frobenius Splitting of Thick Flag Manifolds of Kac–Moody Algebras

2018 ◽  
Vol 2020 (17) ◽  
pp. 5401-5427 ◽  
Author(s):  
Syu Kato
Keyword(s):  
Line Bundle ◽  
Schubert Variety ◽  
Flag Manifold ◽  
Schubert Varieties ◽  
Flag Manifolds ◽  
Flag Varieties ◽  
Moody Algebra ◽  
Frobenius Splitting ◽  
Projective Normality ◽  
Plücker Relations

Abstract We explain that the Plücker relations provide the defining equations of the thick flag manifold associated to a Kac–Moody algebra. This naturally transplants the result of Kumar–Mathieu–Schwede about the Frobenius splitting of thin flag varieties to the thick case. As a consequence, we provide a description of the space of global sections of a line bundle of a thick Schubert variety as conjectured in Kashiwara–Shimozono [13]. This also yields the existence of a compatible basis of thick Demazure modules and the projective normality of the thick Schubert varieties.

scholarly journals FROBENIUS SPLITTING OF EQUIVARIANT CLOSURES OF REGULAR CONJUGACY CLASSES

2006 ◽  
Vol 93 (3) ◽  
pp. 570-592 ◽  
Author(s):  
JESPER FUNCH THOMSEN
Keyword(s):  
Conjugacy Class ◽  
Algebraic Group ◽  
Complete Intersection ◽  
Regular Element ◽  
Positive Characteristic ◽  
Conjugacy Classes ◽  
Closed Field ◽  
Frobenius Splitting ◽  
Equivariant Embedding ◽  
Simply Connected

Let $G$ denote a connected semisimple and simply connected algebraic group over an algebraically closed field $k$ of positive characteristic and let $g$ denote a regular element of $G$. Let $X$ denote any equivariant embedding of $G$. We prove that the closure of the conjugacy class of $g$ within $X$ is normal and Cohen–Macaulay. Moreover, when $X$ is smooth we prove that this closure is a local complete intersection. As a consequence, the closure of the unipotent variety within $X$ shares the same geometric properties.

Monodromic systems on affine flag manifolds

1994 ◽  
Vol 445 (1923) ◽  
pp. 231-246 ◽  
Keyword(s):  
Positive Characteristic ◽  
Flag Manifold ◽  
Flag Manifolds ◽  
Modular Representations ◽  
Affine Weyl Group ◽  
Dual Affine ◽  
Substantial Progress ◽  
Affine Hecke Algebras ◽  
Field Of Positive Characteristic ◽  
Affine Flag

There has been recently substantial progress in the programme of expressing the characters of irreducible modular representations of a semisimple group over a field of positive characteristic in terms of combinatorics of affine Hecke algebras (see Andersen et al . 1992; Kazhdan & Lusztig 1979). This paper is a further contribution to this programme: I explain why in the non-simply laced case, the ‘dual’ affine Weyl group is needed and why in this case it is necessary to use monodromic systems (certain local systems defined on subvarieties of a line bundle) over an affine flag manifold.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

scholarly journals On a conjecture of Pappas and Rapoport about the standard local model for GL_d

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinakar Muthiah ◽  
Alex Weekes ◽  
Oded Yacobi
Keyword(s):  
Type A ◽  
Positive Answer ◽  
Local Model ◽  
Schubert Varieties ◽  
Shimura Varieties ◽  
Local Models ◽  
Full Generality ◽  
Frobenius Splitting ◽  
Affine Grassmannians ◽  
Main Ideas

AbstractIn their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.

On Schubert varieties in the flag manifold of Sl(n,?)

1987 ◽  
Vol 276 (2) ◽  
pp. 205-224 ◽  
Author(s):  
Kevin M. Ryan
Keyword(s):  
Flag Manifold ◽  
Schubert Varieties

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

The Hilbert series of the irreducible quotient of the polynomial representation of the rational Cherednik algebra of type An−1 in characteristic p for p|n − 1

2021 ◽  
pp. 2250191
Author(s):  
Merrick Cai ◽  
Daniil Kalinov
Keyword(s):  
Hilbert Series ◽  
Positive Characteristic ◽  
Closed Field ◽  
Simple Criterion ◽  
Characteristic P ◽  
Polynomial Representation ◽  
Cherednik Algebra ◽  
Rational Cherednik Algebra ◽  
Polynomial Formula ◽  
Field Of Positive Characteristic

In this paper, we study the irreducible quotient [Formula: see text] of the polynomial representation of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] over an algebraically closed field of positive characteristic [Formula: see text] where [Formula: see text]. In the [Formula: see text] case, for all [Formula: see text] we give a complete description of the polynomials in the maximal proper graded submodule [Formula: see text], the kernel of the contravariant form [Formula: see text], and subsequently find the Hilbert series of the irreducible quotient [Formula: see text]. In the [Formula: see text] case, we give a complete description of the polynomials in [Formula: see text] when the characteristic [Formula: see text] and [Formula: see text] is transcendental over [Formula: see text], and compute the Hilbert series of the irreducible quotient [Formula: see text]. In doing so, we prove a conjecture due to Etingof and Rains completely for [Formula: see text], and also for any [Formula: see text] and [Formula: see text]. Furthermore, for [Formula: see text], we prove a simple criterion to determine whether a given polynomial [Formula: see text] lies in [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text] fixed.

Frobenius Splitting and Cohomology Vanishing for Schubert Varieties

1985 ◽  
Vol 122 (1) ◽  
pp. 27 ◽  
Author(s):  
V. B. Mehta ◽  
A. Ramanathan
Keyword(s):  
Schubert Varieties ◽  
Frobenius Splitting

Frobenius splitting and hyperplane sections of flag manifolds

1997 ◽  
Vol 128 (3) ◽  
pp. 437-442 ◽  
Author(s):  
Niels Lauritzen ◽  
Jesper Funch Thomsen
Keyword(s):  
Flag Manifolds ◽  
Frobenius Splitting ◽  
Hyperplane Sections
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