The Infinitesimal Theory Of Quantum Schur Superalgebras And The BLM Realization Of Integral Quantum Supergroups

In1990, Beilinson, MacPherson and Lusztig gave a geometric realization for the quantum enveloping algebra of general linear Lie algebra which is called BLM realization nowadays. In [34], Q. Fu gave the BLM realization of the infinitesimal quantum group of general linear Lie algebra. In that paper, Q. Fu defined the little quantum Schur algebras and gave the standard bases, monomial bases, and BLM bases of little quantum Schur algebras, and gave a presentation of the positive (negative) part of little quantum Schur algebras and the dimension formula of little quantum Schur algebras. Meanwhile, in [20], A. Cox defined the infinitesimal quantum Schur algebras. In [39], Q. Fu gave a presentation for infinitesimal quantum Schur algebras and discussed the relationship between infinitesimal quantum Schur algebras and little quantum Schur algebras. All the results above are called infinitesimal theory. On the other hand, Q. Fu gave the BLM realization of Lusztig integral form of quantum groups of general linear Lie algebras. Motivated by all the results above, we will give a super version of these results in this thesis.This thesis divides into four chapters. The first chapter gives the BLM bases and monomial bases of quantum Schur superalgebras, and studies the relationship between quantum Schur su-peralgebras and finite linear groups. The second chapter we realize the Lusztig integral form of quantum supergroup in terms of quantum Schur superalgebras. The third chapter studies the rela-tionship between infinitesimal quantum supergroups and little quantum superalgebras. The BLM bases and monomial bases of infinitesimal quantum Schur superalgebras and little quantum Schur superalgebras are described. We also give a weight weight idempotent realization of infinitesimal quantum Schur superalgebras which is similar to that in [34]. In the last chapter, we use the general Schur algebra to study the Krull-Schmidt decomposition of tensor products of costandard modules for modular SL3(k).In the first chapter, we use the BLM realization of quantum supergroup U(m|n) in [35] to give the BLM bases and monomial bases of quantum Schur superalgebras. We also realize quantum Schur superalgebras as endomorphism algebras of certain representation of finite general linear groups which generalizes the result on relationship between quantum Schur algebras and finite general linear groups in [3].In the second chapter, motivated by [35] and [40], we find (?)=Z[v, v-1]-submodule of the product of integral quantum Schur superalgebra. This submodule turned out to be a (?)-subalgebra which is isomorphic to the Lusztig (?)-form U of quantum supergroup U(m|n). The advantage of the integral BLM realization is that we can give the BLM realization over any (?)-algebra. For example, we give a BLM of quantum supergroup at a odd root of unity.In the third chapter, motivated by [34], we define the infinitesimal quantum Schur superalgebras and little quantum Schur superalgebras and then give BLM bases and monomial bases of them. At the end of this chapter, we use generating function to give the dimension formula for infinitesimal quantum Schur superalgebras and little quantum Schur superalgebras.In the last chapter, we give an algorithm to determine the Krull-Schmidt decomposition of tensor products of costsndard modules with restricted highest weights for modular SL3(k) using the generalized Schur algebras. In this case, all the highest weights of composition factors of tensor products are in a finite subset it of the weight lattice. The set π is turned out be an ideal with respect to the dominance order in weight lattice. Let G be a reductive group. Then the category of G-modules whose composition factors are in it is equivalent to finite dimensional module over the generalized Schur algebra S(π) of π. The advantage of S(π) is that we simplify the calculation without changing the module category.