Full Record

Author | Li, Ge Jr |

Title | Integral Basis Theorem of cyclotomic Khovanov-Lauda-Rouquier Algebras of type A |

URL | http://hdl.handle.net/2123/8844 |

Publication Date | 2012 |

University/Publisher | University of Sydney |

Abstract | The main purpose of this thesis is to prove that the cyclotomic Khovanov-Lauda-Rouquier algebras of type A over Z are free by giving a graded cellular basis of the cyclotomic KLR algebra. We then extend it to obtain a graded cellular basis of the affine KLR algebra, which indicates that the affine KLR algebra is an affine graded cellular algebra. Finally we work with the Jucys-Murphy elements of the cyclotomic Hecke algebras of type A and proved a periodic property of these elements. |

Subjects/Keywords | Representation theory; KLR algebras; Hecke algebras |

Rights | The author retains copyright of this thesis. |

Country of Publication | au |

Record ID | handle:2123/8844 |

Repository | sydney |

Date Retrieved | 2020-10-09 |

Date Indexed | 2020-10-14 |

Issued Date | 2012-12-12 00:00:00 |

Sample Search Hits | Sample Images

…standard expression. For any 1 ≤ k ≤ m, define s = tλ ·sr1 sr2 . . . srk . Then s is a standard λ-tableau.
1.4. Graded cellular basis of *KLR* *algebras* over a field
13
Proof. The proof is trivial by the definition of the standard expression.
1.3.5…

…Graded cellular basis of *KLR* *algebras* over a field
Suppose O is a field, Hu and Mathas [9, Theorem 5.8] have found a homogeneous basis of
Here we give an equivalent definition of their basis. For any multicomposition λ, recall
t to be the unique…

…Rn (Z)
1.4. Graded cellular basis of *KLR* *algebras* over a field
This section closes with an important Proposition:
Khovanov and Lauda[13][12] have found a basis of Rn (O)
ˆ w | i ∈ I n , w ∈ Sn , 1 , 2…

…1.1. The cyclotomic Khovanov-Lauda-Rouquier *algebras*
5
and relations
(1.1.2)
e
ˆ (i)ˆ
e(j) = δij e
ˆ (i),
(1.1.3)
y
ˆr e
ˆ (i) = e
ˆ (i)yr ,
ˆ (i)
i∈I n e
= 1,
ˆ…

…x28;1.1.2)–(1.1.9). In more detail, if D1 and D2 are two diagrams
6
1. Khovanov-Lauda-Rouquier *Algebras*
then the diagrammatic analogue of the relation e(i)e(j) = δij e(i) is
D1
D1
i1
in
j1
D1 · D2 =
i2…

…consequences of the relations are the following:
1.1. The cyclotomic Khovanov-Lauda-Rouquier *algebras*
i
i
i
i
i
=−
(1.1.15)
i
−
i
i
i
i
i
−
i
i
(1.1.10)
=
(1.1.17)
i
i
i
−
i
i
−
i
=−
(1.1.16)…

…We can now define the main object of study in this thesis,
8
1. Khovanov-Lauda-Rouquier *Algebras*
the cyclotomic Khovanov-Lauda-Rouquier *algebras*, which were introduced by Khovanov and
Lauda [13, Section 3.4].
1.1.18. Definition. The…

…cyclotomic Khovanov-Lauda-Rouquier *algebras* of weight Λ and type
Λ
Λ
Γe is the algebra Rn
(O) = Rn (O)/Nn
(O).
Λ
Λ
Λ
ˆ s + Nn
Therefore, if we write e(i) = e
ˆ (i) + Nn
(O), yr = y
ˆ r + Nn
(O…