On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution

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

Im, Mee Seong

关键词: 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;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/49392/Mee%20Seong_Im.pdf?sequence=1&isAllowed=y

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来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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.