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Articles 1 - 9 of 9
Full-Text Articles in Algebra
Semi-Infinite Flags And Zastava Spaces, Andreas Hayash
Semi-Infinite Flags And Zastava Spaces, Andreas Hayash
Doctoral Dissertations
ABSTRACT SEMI-INFINITE FLAGS AND ZASTAVA SPACES SEPTEMBER 2023 ANDREAS HAYASH, B.A., HAMPSHIRE COLLEGE M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D, UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Ivan Mirković We give an interpretation of Dennis Gaitsgory’s semi-infinite intersection cohomol- ogy sheaf associated to a semisimple simply-connected algebraic group in terms of finite-dimensional geometry. Specifically, we construct machinery to build factoriza- tion spaces over the Ran space from factorization spaces over the configuration space, and show that under this procedure the compactified Zastava space is sent to the support of the semi-infinite intersection cohomology sheaf in the Beilinson-Drinfeld Grassmannian. We also construct …
General Covariance With Stacks And The Batalin-Vilkovisky Formalism, Filip Dul
General Covariance With Stacks And The Batalin-Vilkovisky Formalism, Filip Dul
Doctoral Dissertations
In this thesis we develop a formulation of general covariance, an essential property for many field theories on curved spacetimes, using the language of stacks and the Batalin-Vilkovisky formalism. We survey the theory of stacks, both from a global and formal perspective, and consider the key example in our work: the moduli stack of metrics modulo diffeomorphism. This is then coupled to the Batalin-Vilkovisky formalism–a formulation of field theory motivated by developments in derived geometry–to describe the associated equivariant observables of a theory and to recover and generalize results regarding current conservation.
Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, Matthew Bates
Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, Matthew Bates
Doctoral Dissertations
Let k be the finite field with q elements, let F be the field of Laurent series in the variable 1/t with coefficients in k, and let A be the polynomial ring in the variable t with coefficients in k. Let SLn(F) be the ring of nxn-matrices with entries in F, and determinant 1. Given a polynomial g in A, let Gamma(g) subset SLn(F) be the full congruence subgroup of level g. In this thesis we examine the action of Gamma(g) on the Bruhat-Tits building Xn associated to SLn(F) for n equals 2 and n equals 3. Our first main …
A Relation Between Mirkovic-Vilonen Cycles And Modules Over Preprojective Algebra Of Dynkin Quiver Of Type Ade, Zhijie Dong
A Relation Between Mirkovic-Vilonen Cycles And Modules Over Preprojective Algebra Of Dynkin Quiver Of Type Ade, Zhijie Dong
Doctoral Dissertations
The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian to a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. The relation between this conjecture and the reduceness …
Equations For Nilpotent Varieties And Their Intersections With Slodowy Slices, Benjamin Johnson
Equations For Nilpotent Varieties And Their Intersections With Slodowy Slices, Benjamin Johnson
Doctoral Dissertations
This thesis investigates minimal generating sets of ideals defining certain nilpotent varieties in simple complex Lie algebras. A minimal generating set of invariants for the whole nilpotent cone is known due to Kostant. Broer determined a minimal generating set for the subregular nilpotent variety in all simple Lie algebra types. I extend Broer's results to two families of nilpotent varieties, valid in any simple Lie algebra, that include the nilpotent cone, the subregular case, and usually more. In the first part of my thesis I describe a minimal generating set for the ideal of each of these varieties in the …
Equivariant Intersection Cohomology Of Bxb Orbit Closures In The Wonderful Compactification Of A Group, Stephen Oloo
Equivariant Intersection Cohomology Of Bxb Orbit Closures In The Wonderful Compactification Of A Group, Stephen Oloo
Doctoral Dissertations
This thesis studies the topology of a particularly nice compactification that exists for semisimple adjoint algebraic groups: the wonderful compactification. The compactifica- tion is equivariant, extending the left and right action of the group on itself, and we focus on the local and global topology of the closures of Borel orbits. It is natural to study the topology of these orbit closures since the study of the topology of Borel orbit closures in the flag variety (that is, Schubert varieties) has proved to be inter- esting, linking geometry and representation theory since the local intersection cohomology Betti numbers turned out …
Combinatorics Of Equivariant Cohomology: Flags And Regular Nilpotent Hessenberg Varieties, Elizabeth J. Drellich
Combinatorics Of Equivariant Cohomology: Flags And Regular Nilpotent Hessenberg Varieties, Elizabeth J. Drellich
Doctoral Dissertations
The field of Schubert Calculus deals with computations in the cohomology rings of certain algebraic varieties, including flag varieties and Schubert varieties. In the equivariant setting, GKM theory turns multiplication in the cohomology ring of certain varieties into a combinatorial computation. This dissertation uses combinatorial tools, including Billey’s formula, to do Schubert calculus computations in several varieties. First we address do computations in the equivariant cohomology of full and partial flag varieties, the classical spaces in Schubert calculus. We then do computations in the equivariant cohomology of a family of non-classical spaces: regular nilpotent Hessenberg varieties. The final chapter gives …
Kazhdan-Lusztig Cells In Planar Hyperbolic Coxeter Groups And Automata, Mikhail V. Belolipetsky, Paul E. Gunnells, Richard A. Scott
Kazhdan-Lusztig Cells In Planar Hyperbolic Coxeter Groups And Automata, Mikhail V. Belolipetsky, Paul E. Gunnells, Richard A. Scott
Paul Gunnells
Let C be a one- or two-sided Kazhdan–Lusztig cell in a Coxeter group (W, S), and let Red(C) be the set of reduced expressions of all w ∈ C, regarded as a language over the alphabet S. Casselman has conjectured that Red(C) is regular. In this paper, we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon, and show that our conjectures imply Casselman's.
Generalised Burnside Rings, G-Categories And Module Categories, Paul E. Gunnells, Andrew Rose, Dmitriy Rumynin
Generalised Burnside Rings, G-Categories And Module Categories, Paul E. Gunnells, Andrew Rose, Dmitriy Rumynin
Paul Gunnells
This note describes an application of the theory of generalised Burnside rings to algebraic representation theory. Tables of marks are given explicitly for the groups S4 and S5 which are of particular interest in the context of reductive algebraic groups. As an application, the base sets for the nilpotent element F4(a3) are computed.