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We construct an explicit isomorphism between blocks of cyclotomic Heckealgebras and (sign-modified) Khovanov-Lauda algebras in type A. Theseisomorphisms connect the categorification conjecture of Khovanov and Lauda toAriki's categorification theorem. The Khovanov-Lauda algebras are naturallygraded, which allows us to exhibit a non-trivial Z-grading on blocks ofcyclotomic Hecke algebras, including symmetric groups in positivecharacteristic.

Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras