Classification Of Irreducible Modules For Quantum Schur Superalgebras And A Realization Of Quantum Supergroup U(gl_m|n)

This thesis studies the quantum supergroup U(glm|n)([84]) and its finite dimensional quotient algebras—the quantum Schur superalgebras SF(m|n, r)([37]).On the one hand, we give a complete classification, up to isomorphism, of the irreducible SF(m|n, r)-modules, under the assumption that the ground field has characteristic0, the quantum parameter q∈F is an lth primitive root of unity with l odd, and m+n≥r(Corollary2.4.3). We follow the spirit of Dipper and Du in [25]. At first, we introduce relative norms in SF(m|n,r) as in equation (1.10). Then, we use the symmetric sign q-permutation action of Iwahori-Hecke algebra H((?)r) over tensor space V8(?)r [65], where V is a free module with rank m+n over F, to obtain a basis of SF(m|n,r) consisting of relative norm elements (Theorem1.4.1). The property of the structure constants with respect to the relative norm basis (Theorem1.5.5) yields a filtration of ideals in SF(m|n,r). We also give an alternative descriptions of these ideals. Secondly, we define the Brauer homomorphism φr associated with a pair of positive integers r (r-1, r0) with r-1+lr0=r (Definition2.2.4), and prove that the kernel of φr is the ideal I(Pr0,r) in the above filtration. We define the defect group of a primitive idempotent e∈SF(m|n,r) and show that the indecomposable H(?)r)-module V(?)r e is projective if the defect group of e is trivial. Then, according to the property of the images of the idempotents under the Brauer homomorphism (Theorem2.3.7), we transfer this classification problem into the classifications of irreducible modules of H((?)r and Schur algebras (Lemma2.4.1). As a result, we determine the equivalent classes of idempotents whose images have trivial defect group over Brauer homomorphisms. At last, we achieve the classification of irreducible representations of SF(m|n, r) with m+n≥r at a root of unity, and we also find a bijection between our index set of irreducible modules of SF(m|n,r) and the index set of irreducible modules of Schur superalgebra over a field with characteristic p>0obtained in [15] when l=p.On the other hand, we realize the quantum supergroup U(glm|n) as the "limit" of the quantum Schur superalgebras in the sense that it becomes a subalgebra of the direct product ΠrSF(m|n,r) (Theorem4.4.4). Using the non-symmetric action sign q-permuation action of Iwahori-Hecke alge-bra H((?)r) over tensor space V(?)r[65], we obtain another basis of SF(m|n, r) consisting of relative norm elements which is equivalent to the one defined above. Through a purely algebraic approach based on Algorithms3.1.1,3.1.2and their generalization (Lemma3.1.4) and with a careful analysis to the actions of the products of some relative norm basis elements, we get the super version of two key multiplication formulas (Lemma3.2.1) and their more general forms (Theorem3.3.2) in BLM realization of gln in [6]. We also find an important formula (3.63), which generalizes a formula on orbit dimensions in non-super case, and consider the normalization of the above multiplication formulas (Propositions3.4.2and3.4.3). With all these frameworks, we construct a unified index set for spanning sets of quantum Schur superalgebras (Lemma4.1.3), and then consider, in (4.80), the direct product S(m|n) of quantum Schur superalgebras and its subspace2l(m|n) spanned by a basis with the same index set. The multiplication formula of certain basis elements in the sub-space is obtained (Proposition4.1.6). In this way, we arrive at an algebra homomorphism from U(glm|n) to S(m n)(Theorem4.2.3). We prove this is a monomorphism with2l(m|n) as its image, completing the realization of quantum supergroup U(glm|n)(Theorem4.4.4).