scholarly journals CRITICAL POINTS OF MASTER FUNCTIONS AND FLAG VARIETIES

2004 ◽  
Vol 06 (01) ◽  
pp. 111-163 ◽  
Author(s):  
E. MUKHIN ◽  
A. VARCHENKO
Keyword(s):  
Critical Points ◽  
Flag Variety ◽  
Flag Varieties ◽  
Moody Algebra ◽  
Dominant Weights ◽  
Intersection Points ◽  
One To One ◽  
Schubert Cycles

We consider critical points of master functions associated with integral dominant weights of Kac–Moody algebras and introduce a generating procedure constructing new critical points starting from a given one. The set of all critical points constructed from a given one is called a population. We formulate a conjecture that a population is isomorphic to the flag variety of the Langlands dual Kac–Moody algebra and prove the conjecture for algebras slN+1, so2N+1, and sp2N. We show that populations associated with a collection of integral dominant slN+1-weights are in one to one correspondence with intersection points of suitable Schubert cycles in a Grassmannian variety.

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 LEAF-WISE INTERSECTIONS AND RABINOWITZ FLOER HOMOLOGY

2010 ◽  
Vol 02 (01) ◽  
pp. 77-98 ◽  
Author(s):  
PETER ALBERS ◽  
URS FRAUENFELDER
Keyword(s):  
Critical Points ◽  
Floer Homology ◽  
Symplectic Manifolds ◽  
Action Functional ◽  
Multiplicity Results ◽  
Contact Type ◽  
Existence And Multiplicity ◽  
Rabinowitz Floer Homology ◽  
Intersection Points ◽  
Rabinowitz Action Functional

In this paper we explain how critical points of a particular perturbation of the Rabinowitz action functional give rise to leaf-wise intersection points in hypersurfaces of restricted contact type. This is used to derive existence and multiplicity results for leaf-wise intersection points in hypersurfaces of restricted contact type in general exact symplectic manifolds. The notion of leaf-wise intersection points was introduced by Moser [16].

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.

The double cusp has five minima

1978 ◽  
Vol 84 (3) ◽  
pp. 537-538 ◽  
Author(s):  
J. Callahan
Keyword(s):  
Critical Points ◽  
Standard Form ◽  
Singularity Theory ◽  
The Other ◽  
Universal Unfolding ◽  
The Real ◽  
Level Curves ◽  
Intersection Points ◽  
The One ◽  
Image Position

The double cusp is the real, compact, unimodal singularitysee (2), (4). Functions in a universal unfolding of the double cusp can have nine non-degenerate critical points near the origin, but no more. Index considerations show that precisely four of the nine are saddles, and it has long been part of the folklore of singularity theory that one of the other five must be a maximum. Indeed, a standard form of the unfolded double cusp (1), (3) is a function having a pair of intersecting ellipses as one of its level curves; see Fig. 1(a). There are saddles at the four intersection points, a maximum inside the central quadrilateral, and a minimum inside each of the other four finite regions bounded by the ellipses. The rest of Fig. 1 suggests, however, that a deformation of this function (in which one of the saddles drops below the level of the other three) might turn the maximum into a fifth minimum. The following proposition shows that a function similar to the one in Fig. 1(d) can be realized in an unfolding of the double cusp.

scholarly journals Constructing projective varieties in weighted flag varieties II

2015 ◽  
Vol 158 (2) ◽  
pp. 193-209 ◽  
Author(s):  
MUHAMMAD IMRAN QURESHI
Keyword(s):  
Linear Group ◽  
Lie Group ◽  
General Linear Group ◽  
Hilbert Series ◽  
Flag Variety ◽  
General Linear ◽  
Flag Varieties ◽  
Projective Varieties

AbstractWe give the construction of weighted Lagrangian GrassmannianswLGr(3,6) and weighted partialA3flag varietywFL1,3coming from the symplectic Lie group Sp(6, ℂ) and the general linear group GL(4, ℂ) respectively. We give general formulas for their Hilbert series in terms of Lie theoretic data. We use them as key varieties (Format) to construct some families of polarized 3-folds in codimension 7 and 9. Finally, we list all the distinct weighted flag varieties in codimension (4 ⩽c⩽ 10.

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 Symplectic Degenerate Flag Varieties

2014 ◽  
Vol 66 (6) ◽  
pp. 1250-1286 ◽  
Author(s):  
Evgeny Feigin ◽  
Michael Finkelberg ◽  
Peter Littelmann
Keyword(s):  
Borel Subgroup ◽  
Classical Case ◽  
Explicit Construction ◽  
Flag Variety ◽  
Unipotent Group ◽  
Flag Varieties ◽  
Highest Weight ◽  
Weil Theorem ◽  
Degenerate Flag Varieties ◽  
Pbw Filtration

AbstractA simple finite dimensional module Vλ of a simple complex algebraic group G is naturally endowed with a filtration induced by the PBW-filtration of U(Lie G). The associated graded space is a module for the group Ga, which can be roughly described as a semi-direct product of a Borel subgroup of G and a large commutative unipotent group . In analogy to the flag variety ℱλ = G:[vλ] ⊂ ℙ(Vλ), we call the closure of the Ga-orbit through the highest weight line the degenerate flag variety . In general this is a singular variety, but we conjecture that it has many nice properties similar to that of Schubert varieties. In this paper we consider the case of G being the symplectic group. The symplectic case is important for the conjecture because it is the first known case where, even for fundamental weights ω, the varieties differ from Fω. We give an explicit construction of the varieties and construct desingularizations, similar to the Bott–Samelson resolutions in the classical case. We prove that are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel–Weil theorem and obtain a q-character formula for the characters of irreducible Sp2n-modules via the Atiyah–Bott–Lefschetz fixed points formula.

scholarly journals Four-dimensional Fano quiver flag zero loci

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180791
Author(s):  
Elana Kalashnikov
Keyword(s):  
Electronic Supplementary Material ◽  
Vector Bundles ◽  
Flag Variety ◽  
Flag Varieties ◽  
Joint Work ◽  
Genus Zero ◽  
Fano Manifolds ◽  
Ample Cone ◽  
Quiver Flag Variety ◽  
Supplementary Material

Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of representation theoretic vector bundles. We prove the Abelian/non-Abelian correspondence in this context: this allows us to compute genus zero Gromov–Witten invariants of quiver flag zero loci. We determine the ample cone of a quiver flag variety, and disprove a conjecture of Craw. In the appendices (which can be found in the electronic supplementary material), which are joint work with Tom Coates and Alexander Kasprzyk, we use these results to find four-dimensional Fano manifolds that occur as quiver flag zero loci in ambient spaces of dimension up to 8, and compute their quantum periods. In this way, we find at least 141 new four-dimensional Fano manifolds.

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