University of Sydney Algebra Seminar

Ge Li (University of Sydney)

Friday 16th November, 12:05-12:55pm, Carslaw 173

An integral basis theorem for cyclotomic KLR algebras of type \(A\)

Khovanov and Lauda and Rouquier have introduced a remarkable new family of algebras \(R_n\), the quiver Hecke algebras, for each oriented quiver. The algebras \(R_n\) are naturally \(\mathbb{Z}\)-graded. Brundan and Kleshchev proved that over a field \(F\), the cyclotomic Khovanov-Lauda-Rouquier algebras \(R_n^\Lambda\) are isomorphic to the cyclotomic Hecke algebras of type \(A\), \(H_n^\Lambda\) by constructing an explicit isomorphic mapping, which gives a \(\mathbb{Z}\)-grading to the cyclotomic Hecke algebras. Based on Brundan and Kleshchev's work, Hu and Mathas constructed a graded cellular basis with some restriction. In this talk I will show that such restriction can be removed and the graded cellular basis introduced by Hu and Mathas can be extended to \(R_n^\Lambda\) over \(\mathbb{Z}\). Furthermore we will show that the graded cellular basis can be extended to affine Khovanov-Lauda-Rouquier algebras and it gives a classification of all simple \(R_n\)-modules.