Tight Monomials In Quantum Group For Type D₄

Canonical basis plays an important role in quantum groups and their representa-tions, and is a powerful tool to study quantum groups and their representations, one has done one's best to determine elements in Canonical basis explicitly. But so far, elements in Canonical basis for only few quantum groups have been computed com-pletely, in which the key step is to find monomials in Canonical basis(so called tight monomials). All of tight monomials with t-value≤8in quantum group for type D4are gained in this paper.The paper is divided into five Chapters, and is organized as follows:In Chapter1, the background of quantum group and Canonical basis in quantum group is introduced, and the motivation of this paper is furnished.In Chapter2, the definition of quantum group and some related concepts, also algebraic definition of Canonical basis of quantum group by Lusztig are recalled.In Chapter3, the known results about Canonical basis of quantum groups and monomials in quantum groups are listed.In Chapter4, based on introduction to quiver, Cartan datum, etc, all of tight monomials with t-value≤8in quantum group for type D4are completely given.In Chapter5, the main results in Chapter4are proved.