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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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…related to categorification of quantum groups ([KL09], [KL11], [Rou08], [Bru13]), graded cyclotomic q-Schur algebras as a quotient of a convolution algebra in the study of quiver varieties via a subset of…

…Q1 . There is a natural action of the group Gβ := GLβi (C) on Rep(Q, β) where each factor GLβi (C) acts via change i∈Q0 of basis in Cβi . We say a quiver is an (affine) Dynkin quiver if the underlying graph…

…has the structure of an (affine) Dynkin graph. We say a quiver is a k-Kronecker quiver if the quiver has two vertices and exactly k arrows (of any direction) between them, and we say a quiver is an m-Jordan quiver if the quiver has…

…the results presented here, consider ADE-Dynkin quivers and affine type Ar -Dynkin quivers. If Q if an ADE-Dynkin quiver, we will write r to mean Q has r vertices and r − 1 arrows. Let Pβ = Pi be a product of parabolic groups as defined above. We will…

…then C[F • Rep(Q, β)]Pβ ,χ ∼ = C[tn⊕r−1 ]. χ 0 2. If Q is an affine type Ar -Dynkin quiver and β = (n, . . . , n) ∈ ZQ ≥0 , then C[F • Rep(Q, β)]Pβ ,χ ∼ ]. = C[t⊕r+1 n χ More…

…We say Q is of affine (type) Ar -Dynkin if the underlying graph is the affine Ar -Dynkin graph: r+1 • 5 • ... 4 1 •. • 2 3 • • Note that A0 has one vertex and one arrow whose head equals its tail, and A1 has two vertices joined…

…pair (Q, I), where Q is a quiver and I is a two-sided ideal of CQ generated by relations. The quotient algebra CQ/I is the path algebra of (Q, I). Example 2.1.19. Let Q be an A3 -Dynkin quiver given in Example 2.1.14 and let I = a2…