scholarly journals Weighted PBW degenerations and tropical flag varieties

2019 ◽  
Vol 21 (01) ◽  
pp. 1850016 ◽  
Author(s):  
X. Fang ◽  
E. Feigin ◽  
G. Fourier ◽  
I. Makhlin
Keyword(s):  
Flag Variety ◽  
Polyhedral Cone ◽  
Flag Varieties ◽  
Type Formula ◽  
Toric Degeneration ◽  
Degree Function ◽  
Maximal Cone ◽  
Partial Flag Varieties

We study algebraic, combinatorial and geometric aspects of weighted Poincaré–Birkhoff–Witt (PBW)-type degenerations of (partial) flag varieties in type [Formula: see text]. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone, which can be identified with a maximal cone in the tropical flag variety. Varying the degree function in the cone, we recover, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration.

scholarly journals Chevalley-Monk and Giambelli formulas for Peterson Varieties

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Elizabeth Drellich
Keyword(s):  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Type A ◽  
Flag Variety ◽  
Flag Varieties ◽  
Dimensional Torus ◽  
One Dimensional ◽  
International Audience ◽  
Schubert Classes ◽  
Partial Flag Varieties

International audience A Peterson variety is a subvariety of the flag variety $G/B$ defined by certain linear conditions. Peterson varieties appear in the construction of the quantum cohomology of partial flag varieties and in applications to the Toda flows. Each Peterson variety has a one-dimensional torus $S^1$ acting on it. We give a basis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ring generators. In type A Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave Giambelli's formula for multiplication in the cohomology ring. This paper gives a Chevalley-Monk rule and Giambelli's formula for all Lie types.

scholarly journals CLOSED ORBITS ON PARTIAL FLAG VARIETIES AND DOUBLE FLAG VARIETY OF FINITE TYPE

2014 ◽  
Vol 68 (1) ◽  
pp. 113-119 ◽  
Author(s):  
Kensuke KONDO ◽  
Kyo NISHIYAMA ◽  
Hiroyuki OCHIAI ◽  
Kenji TANIGUCHI
Keyword(s):  
Finite Type ◽  
Flag Variety ◽  
Flag Varieties ◽  
Closed Orbits ◽  
Partial Flag Varieties

scholarly journals Twisted gamma filtration and algebras with orthogonal involution

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Caroline Junkins
Keyword(s):  
Algebraic Group ◽  
Grothendieck Group ◽  
Linear Algebraic Group ◽  
Flag Variety ◽  
Flag Varieties ◽  
Torsion Element ◽  
Complete Flag ◽  
Gamma Filtration

AbstractFor the Grothendieck group of a split simple linear algebraic group, the twisted γ-filtration provides a useful tool for constructing torsion elements in -rings of twisted flag varieties. In this paper, we construct a non-trivial torsion element in the γ-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline.

Affine flag varieties

2020 ◽  
pp. 191-206
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Group Scheme ◽  
Flag Variety ◽  
Flag Varieties ◽  
Connected Component ◽  
Witt Vector ◽  
Classical Definition ◽  
Definition Of ◽  
Affine Flag Variety ◽  
Affine Flag

This chapter reviews affine flag varieties. It generalizes some of the previous results to the case where G over Zp is a parahoric group scheme. In fact, slightly more generally, it allows the case that the special fiber is not connected, with connected component of the identity G? being a parahoric group scheme. This case comes up naturally in the classical definition of Rapoport-Zink spaces. The chapter first discusses the Witt vector affine flag variety over Fp. This is an increasing union of perfections of quasiprojective varieties along closed immersions. In the case that G° is parahoric, one gets ind-properness.

scholarly journals Relative Equivariant Motives and Modules

2019 ◽  
pp. 1-29
Author(s):  
Baptiste Calmès ◽  
Alexander Neshitov ◽  
Kirill Zainoulline
Keyword(s):  
Generic Version ◽  
Flag Variety ◽  
Flag Varieties ◽  
Direct Sum ◽  
Chow Motives ◽  
Direct Sum Decomposition ◽  
Oriented Cohomology ◽  
Demazure Modules ◽  
Algebraic Cobordism

Abstract We introduce and study various categories of (equivariant) motives of (versal) flag varieties. We relate these categories with certain categories of parabolic (Demazure) modules. We show that the motivic decomposition type of a versal flag variety depends on the direct sum decomposition type of the parabolic module. To do this we use localization techniques of Kostant and Kumar in the context of generalized oriented cohomology as well as the Rost nilpotence principle for algebraic cobordism and its generic version. As an application, we obtain new proofs and examples of indecomposable Chow motives of versal flag varieties.

scholarly journals The Hardy-Littlewood property of flag varieties

2003 ◽  
Vol 170 ◽  
pp. 185-211 ◽  
Author(s):  
Takao Watanabe
Keyword(s):  
Asymptotic Distribution ◽  
Rational Points ◽  
Flag Variety ◽  
Flag Varieties

AbstractWe study the asymptotic distribution of rational points on a generalized flag variety which are of bounded height and satisfy some congruence conditions in the formulation analogous to a strongly Hardy-Littlewood variety.

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