Structure of Quiver Polynomials and Schur Positivity Public Deposited

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Last Modified
  • March 21, 2019
Creator
  • Kaliszewski, Ryan
    • Affiliation: College of Arts and Sciences, Department of Mathematics
Abstract
  • Given a directed graph (quiver) and an association of a natural number to each vertex, one can construct a representation of a Lie group on a vector space. If the underlying, undirected graph of the quiver is a Dynkin graph of A-, D-, or E-type then the action has finitely many orbits. The equivariant fundamental classes of the orbit closures are the key objects of study in this paper. These fundamental classes are polynomials in universal Chern classes of a classifying space so they are referred to as "quiver polynomials." It has been shown by A. Buch [B08] that these polynomials can be expressed in terms of Schur-type functions. Buch further conjectures that in this expression the coefficients are non-negative. Our goal is to study the coefficients and structure of these quiver polynomials using an iterated residue description due to R. Rimanyi [RR]. We introduce the Jacobi-Trudi transform, which creates an equivalence realtion on rational functions, to show that Buch's conjecture holds for a quiver polynomial if and only if there is a representative in the equivalence class that is Schur positive. Also we define a notion of strong Schur positivity and demonstrate the connection between this and Schur positivity, proving Schur positivity for some special cases of quiver polynomials.
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  • In Copyright
Advisor
  • Rimanyi, Richard
Degree
  • Doctor of Philosophy
Degree granting institution
  • University of North Carolina at Chapel Hill
Graduation year
  • 2013
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