Full Record

Author | Loubert, Joseph |

Title | Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A |

URL | http://hdl.handle.net/1794/19255 |

Publication Date | 2015 |

Date Accessioned | 2015-08-18 23:02:53 |

Degree | PhD |

Discipline/Department | Department of Mathematics |

Degree Level | doctoral |

University/Publisher | University of Oregon |

Abstract | This thesis consists of two parts. In the first we prove that the Khovanov-Lauda-Rouquier algebras $R_\alpha$ of finite type are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in $R_\alpha$ are generated by idempotents. This in particular implies the (known) result that the global dimension of $R_\alpha$ is finite. In the second part we use the presentation of the Specht modules given by Kleshchev-Mathas-Ram to derive results about Specht modules. In particular, we determine all homomorphisms from an arbitrary Specht module to a fixed Specht module corresponding to any hook partition. Along the way, we give a complete description of the action of the standard KLR generators on the hook Specht module. This work generalizes a result of James. This dissertation includes previously published coauthored material. |

Subjects/Keywords | Affine cellularity; KLR algebras; Specht modules |

Contributors | Kleshchev, Alexander (advisor) |

Language | en |

Rights | Creative Commons BY 4.0-US [Always confirm rights and permissions with the source record.] |

Country of Publication | us |

Record ID | handle:1794/19255 |

Repository | oregon |

Date Retrieved | 2020-06-15 |

Date Indexed | 2020-06-18 |

Grantor | University of Oregon |

Issued Date | 2015-08-18 00:00:00 |

Sample Search Hits | Sample Images | Cited Works

…and Ram (20) provide a presentation for the Specht modules as modules over the
full *KLR* algebra R, thus redefining these classical objects purely in the context of
the *KLR* *algebras*. Our second main theorem determines Hom(S µ , S λ…

…x29; when λ is a
hook.
Affine Cellularity of *KLR* *Algebras* of Finite Types
The content of chapter II has already been published as (18). The goal of
chapter II is to establish (graded) affine cellularity in the sense of Koenig and Xi…

…graded) cellularity of cyclotomic *KLR* *algebras* of finite types.
Our approach is independent of the homological results in (30), (13) and (3)
(which relies on (30)). The connection between the theory…

…theoretic notation that we employ.
We move on in subsection 2.1 to the definition and basic results of KhovanovLauda-Rouquier (*KLR*) *algebras*. The next two subsections are devoted to recalling
results about the representation theory of *KLR* *algebras*…

…Then, in subsection 2.1,
we introduce our notation regarding quantum groups, and recall some well-known
basis theorems. The next subsection is devoted to the connection between *KLR*
*algebras* and quantum groups, namely the categorification theorems…

…Finally,
subsection 2.1 contains an easy direct proof of a graded dimension formula for the
*KLR* *algebras*, cf. (3, Corollary 3.15).
6
Section 3 is devoted to constructing a basis for the *KLR* *algebras* that is
amenable to checking affine…

…basis in full generality.
In section 4 we show how the affine cellular basis is used to prove that the
*KLR* *algebras* are affine cellular.
Finally, in section 5 we verify Hypothesis 2.2.9 for all positive roots in all
finite types. We begin in subsection…

…RdΛ0 known as a cyclotomic *KLR*
(1)
(Khovanov-Lauda-Rouquier) algebra of type Ae−1 when e 6= 0, and type A∞ when
e = 0. The Specht modules are described in (4) as modules over the cyclotomic
*KLR* *algebras*. This result is…