The q-analogue of the alternating groups and its representations

Frobenius began the study of representation theory and character theory of the symmetric groups Sn at the turn of the century [4]. In this study, He showed that there is one irreducible representation πλ of Sn corresponding to each partition λ of n and this correspondence gives all of irreducible representations of Sn up to equivalency. After this work, he also began the study of representation theory and character theory of the alternating groups An at the opening of the 20th century [5]. In this study, he showed that there are branching rules when we restrict irreducible representation of the Sn to An. We denote by π̃λ the restriction of πλ to An . If λ is self-conjugate(that is symmetric figure as Young diagram along its diagonal), then π̃λ is decomposed into the direct sum of two irreducible representations whose degrees are coincide. But these two representations are mutually inequivalent. On the other hand, if λ is non self-conjugate, then π̃λ is also the irreducible and π̃λ is equivalent to π̃tλ where λ is transpose of λ. These representations consist all of irreducible representations of An up to equivalency(Theorem 3.4). The concrete construction of irreducible representations of Sn were given by Young. He gave irreducible representations of Sn in two ways: ‘seminormal form’, and ‘orthogonal form’ (For detail, see [2]). The Hecke algebra Hn(q) of type A is considered as the q-analogue of the group ring C[Sn] of the symmetric group Sn, and its irreducible representations are constructed as well as in the case of Sn under some assumptions about the value of the parameter q. The seminormal form for Hn(q) was given by Hoefsmit [9], and the orthogonal form was given by Wenzl [15]. It was not seem to be seen about the q-analogue of the group ring C[An] of the alternating group An so far. In this context, we defined a subalgebra An(q) of Hn(q) whose dimension as vector space over C is just a half of that of Hn(q). The group ring C[An] which is generated by following elements,