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Springer theory and the geometry of quiver flag varieties

Sauter, Julia Anneliese (2013) Springer theory and the geometry of quiver flag varieties. PhD thesis, University of Leeds.

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Abstract

1. Springer theory. For any projective map E ! V , Chriss and Ginzburg defined an algebra structure on the (Borel-Moore) homology Z := H_(E _V E), which we call Steinberg algebra. (Graded) Projective and simple Z-modules are controlled by the BBD-decomposition associated to E ! V . We restrict to collapsings of unions of homogeneous vector bundles over homogeneous spaces because we have the cellular fibration technique and for equivariant Borel-Moore homology we can use localization to torus-fixed points. Examples of Steinberg algebras include group rings of Weyl groups, Khovanov-Lauda-Rouquier algebras, nil Hecke algebras. 2. Steinberg algebras. We choose a class of Steinberg algebras and give generators and relations for them. This fails if the homogeneous spaces are partial and not complete flag varieties, we call this the parabolic case. 3. The parabolic case. In the parabolic cases, we realize the Steinberg algebra ZP as corner algebra in a Steinberg algebra ZB associated to Borel groups (this means ZP = eZBe for an idempotent element e 2 ZB). 4. Monoidal categories. We explain how to construct monoidal categories from families of collapsings of homogeneous bundles. 5. Construct collapsings. We construct collapsing maps over given loci which are resolutions of singularities or generic Galois coverings. For closures of homogeneous decomposition classes of the Kronecker quiver these maps are new. 6. Quiver flag varieties. Quiver flag varieties are the fibres of certain collapsings of homogeneous bundles. We investigate when quiver flag varieties have only finitely many orbits and we describe the category of flags of quiver representations as a _-filtered subcategory for the quasi-hereditary algebra KQ KAn. 7. An-equioriented. For the An-equioriented quiver we find a cell decompositions of the quiver flag varieties, which are parametrized by certain multi-tableaux.

Item Type: Thesis (PhD)
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds)
Identification Number/EthosID: uk.bl.ethos.605354
Depositing User: Repository Administrator
Date Deposited: 12 Jun 2014 08:41
Last Modified: 03 Sep 2014 10:49
URI: http://etheses.whiterose.ac.uk/id/eprint/6318

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