𝐺-stable pieces and partial flag varieties

2009 ◽  
pp. 61-70 ◽  
Author(s):  
Xuhua He
Keyword(s):  
Flag Varieties ◽  
Partial Flag Varieties

scholarly journals Braiding via geometric Lie algebra actions

2012 ◽  
Vol 148 (2) ◽  
pp. 464-506 ◽  
Author(s):  
Sabin Cautis ◽  
Joel Kamnitzer
Keyword(s):  
Lie Algebra ◽  
Braid Group ◽  
Group Actions ◽  
Derived Categories ◽  
Flag Varieties ◽  
Coherent Sheaves ◽  
Cotangent Bundles ◽  
Partial Flag Varieties

AbstractWe introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows that strong categorical actions in the sense of Khovanov–Lauda and Rouquier also lead to braid group actions. As an example, we construct an action of Artin’s braid group on derived categories of coherent sheaves on cotangent bundles to partial flag varieties.

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 Moduli of toric vector bundles

2008 ◽  
Vol 144 (5) ◽  
pp. 1199-1213 ◽  
Author(s):  
Sam Payne
Keyword(s):  
Linear Group ◽  
Group Action ◽  
Chern Class ◽  
Vector Bundles ◽  
Closed Subscheme ◽  
Flag Varieties ◽  
Moduli Stack ◽  
Rank Conditions ◽  
Partial Flag Varieties

AbstractWe give a presentation of the moduli stack of toric vector bundles with fixed equivariant total Chern class as a quotient of a fine moduli scheme of framed bundles by a linear group action. This fine moduli scheme is described explicitly as a locally closed subscheme of a product of partial flag varieties cut out by combinatorially specified rank conditions. We use this description to show that the moduli of rank three toric vector bundles satisfy Murphy’s law, in the sense of Vakil. The preliminary sections of the paper give a self-contained introduction to Klyachko’s classification of toric vector bundles.

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