We introduce a new family of superalgebras which should be considered as a super version of the Khovanov-Lauda-Rouquier algebras. Let I be the set of vertices of a Dynkin diagram with a decomposition I = I even ⊔ I odd . To this data, we associate a family of graded superalgebras R n , the quiver Hecke superalgebras. When I odd = ∅, these algebras are nothing but the usual Khovanov-Lauda-Rouquier algebras. We then define another family of graded superalgebras RC n , the quiver Hecke-Clifford superalgebras, and show that the superalgebras R n and RC n are weakly Morita superequivalent to each other. Moreover, we prove that the affine Hecke-Clifford superalgebras, as well as their degenerate version, the affine Sergeev superalgebras, are isomorphic to quiver Hecke-Clifford superalgebras RC n after a completion.2000 Mathematics Subject Classification. Primary 81R50, Secondary 20C08.

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Abstract. We give the graded Cartan determinants of the symmetric groups. Based on it, we propose a gradation of Hill's conjecture which is equivalent to Külshammer-OlssonRobinson's conjecture on the generalized Cartan invariants of the symmetric groups.