A Presentation Of The Little Q-Schur Algebra Uq(2, R) At The Third Root Of Unity

The little q-Schur algebra uq(n,r) was defined and its structure was investigated by Du Jie, Fu Qiang and Wang Jianpan in [20]. In particular, various bases for the algebra were constructed. In this paper, we will study the realization of the little q-Schur algebra uq(2,r) when the parameter q is the third root of unity by giving generators and relations. That is, we consider the presentation problem of the algebra. According to the value of r, we divide the task into several cases. For each case, we define an algebra by generators and relations, and then prove that the algebra is isomorphic to the given little q-Schur algebra. To establish the isomorphism, known results on the BLM basis and on the coordinate algebra, as well as those on the polynomial representations of q-Schur algebras, are used to calculate the multiplication table with respect to the BLM basis for the little q-Schur algebra. From the table the required isomorphism is easy to be seen, and therefore, a presentation of the little q-Schur algebra follows. Moreover, our approach also applies to other odd roots of unity for certain special q. We also do the generalizations.