Comparing Generalized Schur Algebras And The Double Centeralizer Property

This PhD dissertation is on the polynomial representations of the general linear group GL_n and related topics. It consists of three main parts.1. We construct a large family of isomorphisms between the quasi-hereditary quotients of the quantized Schur algebras. By using these isomorphisms, we obtain factorizations of the maps constructed by Beilinson-Lusztig-MacPherson; reprove Jame's column (resp. row) removal algorithms for decomposition numbers; and discover the algebra isomorphisms hidden behind these algorithms; obtain the column (resp. row) removal algorithms for p-Kostka numbers, which are known only for column removal, and for row removal in some special cases. On the other hand, these isomorphisms induce a family of equivalences between the abelian full subcategories of different degrees of the module category of quantum GL_n, from which we get the invariance of the subcategories under the translation. By setting q = 1, we then obtain the corresponding results for the Schur algebras, which are still new.The module category of quantum GL_n is a direct sum of the ones of infinitely many q-Schur algebras. The family of isomorphisms between the quasi-hereditary quotients of the quantized Schur algebras we constructed reflects the "local to global" principle in the study of quantum GL_n.2. We construct and prove a new straightening formula of bi-determinants. Different from the existed straightening formulas, the new one is a complete version in the sense that it contains not only the terms of the highest weight but also the terms of lower weights. Besides, we obtain an important property of this new formula: it is compatible with the complement construction. As an application, we prove that those isomorphisms above for generalized Schur algebras over arbitrary infinite fields actually, also hold for generalized Schur algebras over arbitrary commutative rings; moreover, these isomorphisms can be expressed explicitly with respect to the dual basis of the semi-standard bi-determinants.3. We introduce a new and special class of quasi-hereditary algebras A_q, and prove the Hemmer-Nakano equivalences for them provided that q is sufficiently large. Such equivalences were known before only for q-Schur algebras.