The q-Analogue of the Alternating Group and Its Representations

We define the new algebra. This algebra has a parameter q. The defining relations of this algebra at q = 1 coincide with the basic relations of the alternating group. We also give the new subalgebra of the Hecke algebra of type A which is isomorphic to this algebra. This algebra is free of rank half that of the Hecke algebra. Hence this algebra is regarded as a q-analogue of the alternating group.All the isomorphism classes of the irreducible representations of this algebra and the q-analogue of the branching rule between the symmetric group and the alternating group are obtained.