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Author
Title Integral Basis Theorem of cyclotomic Khovanov-Lauda-Rouquier Algebras of type A
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Publication Date
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
Date Indexed 2020-10-14
Issued Date 2012-12-12 00:00:00

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…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…

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