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We define a quotient of the category of finitely generated modules over thecyclotomic Khovanov-Lauda-Rouquier algebra for type An and show it has a modulecategory structure over a direct sum of certain cyclotomicKhovanov-Lauda-Rouquier algebras of type An-1, this way categorifying thebranching rules for the inclusion of sl(n) in sl(n+1). Using this we give anew, elementary proof of Khovanov-Lauda cyclotomic conjecture. We show thatcontinuing recursively gives the Gelfand-Tsetlin basis for type An. As anapplication we prove a conjecture of Mackaay, Stosic and Vaz concerningcategorical Weyl modules.

KLR algebras and the branching rule I: the categorical Gelfand-Tsetlin basis in type An