Study On Hecke Algebra And Related Algebras

The main results of this paper are divided into four parts.Firstly, we use the topological results of Grassmannian manifold to discuss the ba-sic algebra’s cellular basis of Cyclotomic NilHecke algebra NHl,n.This basis was found originally by Jun Hu, when he was researching a complete primitive idempotent ele-ments. In this part of our work, we established an isomorphism between the basic alge-bra of Cyclotomic NilHecke algebra and the algebra formed by cohomology of Grass-mannian manifold. Furthermore, we constructed a more specific isomorphism, which give a precise one-to-one correspondence between the basis given by Hu and Schubert cells in Grasssmanian. With help of this isomorphism, we can recover some classic con-sequence in Grassmanian then use them to explain some algebraic problems in NHl,n. For example, we can prove easily that the basis given by Jun Hu is a graded cellular basis, which make our basic algebra to be a graded cellular algebra.Secondly, we discuss some problems in Iwahori-Hecke algebra, other related alge-bras and their module categories. We began with the borel subalgebra of q-Schur alge-bra, and described the maximal ideal of this algebra. By using this consequences, we achieved a complex in borel subalgebra category by bar-resolution method. After Induced Funtor and Schur Funtor, we can get a complex on dual Specht module in Iwahori-Hecke algebra’s category. Moreover, combine with results of partial exactness of this complex discovered by Boltje and Maisch. We finally showed that this complex is a projective resolution of dual Specht module.Thirdly, in third part of this paper, different from Weyl modules in other people’s theory such as defined as quotient modules of certain "permutation" module, we re-alized Weyl modules of Cyclotomic Schur algebra as a series of regular modules with help of a group of special elements. Some related conception such as cellular basis we can also realize and prove in this environment. Furthermore, we reprove the branch rule of Cyclotomic Schur algebra by using our new basis.Finally, we show some researching attempts trying in our last part. This part we focus on two problems:First is that we use the projective resolution which was con- structed in second part of our work, to achieve some properties of the extension module between dual Specht modules. We also found a combinatorial way to describe a basis in these modules. In rest of this part, in order to extend the Woodcook condition of Iwahori-Hecke algebra to Cyclotomic Hecke algebra, we tried to establish the theory of global basis in Cyclotomic Schur algebra and its Weyl modules. Then, we can easily tell that Boltje-Maisch projective resolution is still valid in cyclotomical case.