The Linear Monomials In The Subgroups Of B ₃

Quantum group also called quantized enveloping algebra, it has many classes of bases, which the most powerful one is Canonical basis. Canonical basis plays an im-portant role in the theory of the quantum groups and their representations, also is a powerful tool to study quantum groups and their representations, and the simplest elements in Canonical basis are tight monomials. We compute all the tight monomials with7≤t-value≤9in quantum group for type B3in this paper.This paper is divided into five parts. The first part introduces the background and its application case of Canonical basis of quantum group, and reviews research current of Canonical basis, raises problems of this paper. The second part recalls some concepts about quantum group and algebraic definition of Canonical basis of quantum group by G.Lusztig. The third part lists the known results about Canonical basis of quantum group for simply-laced case and non simply laced case. The fourth part suggests quiver and related knowledge,presents tightness criterion of monomials, compute all the tight monomials with7≤t-value≤9in quantum group for type B3. The fifth part proves the results in the fourth part...