Full Record

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 |

Sample Search Hits | Sample Images

…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…