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 Author Im, Mee Seong Title On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution URL http://hdl.handle.net/2142/49392 Publication Date 2014 Date Accessioned 2014-05-30 16:41:46 Degree PhD Discipline/Department 0439 Degree Level doctoral University/Publisher University of Illinois – Urbana-Champaign Abstract We introduce the notion of filtered representations of quivers, which is related to usual quiver representations, but is a systematic generalization of conjugacy classes of $n\times n$ matrices to (block) upper triangular matrices up to conjugation by invertible (block) upper triangular matrices. With this notion in mind, we describe the ring of invariant polynomials for interesting families of quivers, namely, finite $ADE$-Dynkin quivers and affine type $\widetilde{A}$-Dynkin quivers. We then study their relation to an important and fundamental object in representation theory called the Grothendieck-Springer resolution, and we conclude by stating several conjectures, suggesting further research. Subjects/Keywords Algebraic geometry; representation theory; quiver varieties; filtered quiver variety; quiver flag variety; semi-invariant polynomials; invariant subring; Derksen-Weyman; Domokos-Zubkov; Schofield-van den Bergh; ADE-Dynkin quivers; affine Dynkin quivers; quivers with at most two pathways between any two vertices; filtration of vector spaces; classical invariant theory; the Hamiltonian reduction of the cotangent bundle of the enhanced Grothendieck-Springer resolution; almost-commuting varieties; affine quotient Contributors Nevins, Thomas A. (advisor); Kedem, Rinat (Committee Chair); Nevins, Thomas A. (committee member); Bergvelt, Maarten J. (committee member); Schenck, Henry K. (committee member) Language en Rights Copyright 2014 Mee Seong Im Country of Publication us Record ID handle:2142/49392 Repository uiuc Date Indexed 2020-03-09 Grantor University of Illinois at Urbana-Champaign Issued Date 2014-05-30 16:41:46

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…be treated here. Instead, we focus on the most basic algebraic problem about such spaces, namely to describe their function theory. This amounts to a generalization of classical problems of invariant theory to this new context. We begin with some…

…Van den Bergh, and Domokos-Zubkov are based on classical invariant theory of semisimple or reductive groups and do not apply in the filtered situation. As a result, the problem seems to be much harder, and we do not achieve a complete solution here…

…The study of the space F • Rep(Q, β) with its Pβ -action thus represents a generalization of the study of upper triangular linear operators up to upper-triangular change of basis. Next, we introduce some definitions from invariant theory. Let…

…strategy to produce all semi-invariant polynomials for acyclic quivers by using representation theory techniques, and in 2001, Domokos-Zubkov gave a strategy on producing all semi-invariant polynomials for all quiver varieties using combinatorical…

…x28;W (a)), det(W (a))], which coincides with the classical result that generators of the ring of invariant polynomials are precisely the coefficients of the characteristic polynomial of W (a). Example…

…preserves the filtration. It is related to very interesting developments in algebraic geometry and representation theory, in particular, to Khovanov-Lauda and Rouquier algebras (KLR-algebras) which are graded algebras whose representation theory is…

…Gβ -action represents a far-reaching generalization of the classical study of similarity classes of matrices, i.e., the study of linear operators up to change of basis. Now, we introduce a new, refined analogue of quiver representations. Given a…

…g ∈ G and for all x ∈ X, then f is called an invariant polynomial. If f (g · x) = χ(g)f (x) for all g ∈ G and x ∈ X, where χ is a character of G, then f is called a χ-semi-invariant polynomial. Definition 1.0.4. We…