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Cluster Algebras And Maximal Green Sequences For Closed Surfaces, Eric Bucher
Cluster Algebras And Maximal Green Sequences For Closed Surfaces, Eric Bucher
LSU Doctoral Dissertations
Given a marked surface (S,M) we can add arcs to the surface to create a triangulation, T, of that surface. For each triangulation, T, we can associate a cluster algebra. In this paper we will consider orientable surfaces of genus n with two interior marked points and no boundary component. We will construct a specific triangulation of this surface which yields a quiver. Then in the sense of work by Keller we will produce a maximal green sequence for this quiver. Since all finite mutation type cluster algebras can be associated to a surface, with some rare exceptions, this work …
Conical Representations For Direct Limits Of Riemannian Symmetric Spaces., Matthew Glenn Dawson
Conical Representations For Direct Limits Of Riemannian Symmetric Spaces., Matthew Glenn Dawson
LSU Doctoral Dissertations
We extend the definition of conical representations for Riemannian symmetric space to a certain class of infinite-dimensional Riemannian symmetric spaces. Using an infinite-dimensional version of Weyl's Unitary Trick, there is a correspondence between smooth representations of infinite-dimensional noncompact-type Riemannian symmetric spaces and smooth representations of infinite-dimensional compact-type symmetric spaces. We classify all smooth conical representations which are unitary on the compact-type side. Finally, a new class of non-smooth unitary conical representations appears on the compact-type side which has no analogue in the finite-dimensional case. We classify these representations and show how to decompose them into direct integrals of irreducible conical …
Multiplicity Formulas For Perverse Coherent Sheaves On The Nilpotent Cone, Myron Minn-Thu-Aye
Multiplicity Formulas For Perverse Coherent Sheaves On The Nilpotent Cone, Myron Minn-Thu-Aye
LSU Doctoral Dissertations
Arinkin and Bezrukavnikov have given the construction of the category of equivariant perverse coherent sheaves on the nilpotent cone of a complex reductive algebraic group. Bezrukavnikov has shown that this category is in fact weakly quasi-hereditary with Andersen--Jantzen sheaves playing a role analogous to that of Verma modules in category O for a semi-simple Lie algebra. Our goal is to show that the category of perverse coherent sheaves possesses the added structure of a properly stratified category, and to use this structure to give an effective algorithm to compute multiplicities of simple objects in perverse coherent sheaves. The algorithm is …