In a remarkable series of papers Brundan and Kleshchev (and Wang) have shown that that group algebra of the symmetric group, and more generally the cyclotomic Hecke algebras and their rational degenerations, are Z-graded. They then significantly extended Ariki's categorification theorem to compute the corresponding graded decomposition numbers of these algebras in characteristic zero in terms of the canonical bases of higher level Fock spaces. The starting point for this work was Khovanov and Lauda's construction of a family of algebras which categorify the negative part of the quantum group of a Kac-Moody algebra. The first result of Brundan and Kleshchev showed that the cyclotomic Hecke algebras are isomorphic to the Khovanov-Lauda algebras of type A and, consequently, that these algebras are Z-graded.
In this talk I will survey these results and then describe how to construct a graded cellular basis of the cyclotomic Khovanov-Lauda algebras by building on the representation theory of the cyclotomic Hecke algebras. This result is significant because it gives first known homogeneous basis of these algebras. As a consequence we prove several conjectures of Brundan, Kleshchev and Wang.
This is joint work with Jun Hu.
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After the seminar we will be taking the speaker to lunch.
See the Algebra Seminar web page for information about other seminars in the series.
James East [email protected]