Standard Basis Theorem for Quantum Linear Groups

Abstract

A linear basis indexed by standard bitableaux is given for quantum linear semi-groups. As a consequence, one can easily derive a standard basis for quantum linear groups. From a combinatorial point of view, quantum linear semigroups are identical with what results from quantization of the letterplace algebra Rota and co-workers, Adv. Math. 27 (1978), 63-92. The standard basis theorem proved here can be viewed as a quantum straightening formula. The present paper is written in the language of supersymmetric algebra. In doing so, we actually have obtained a standard basis for Manin's quantum linear supersemigroups