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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…