Comparing $GL_n$-Representations by Characteristic-Free Isomorphisms between Generalized Schur Algebras

TL;DR

This paper constructs characteristic-free isomorphisms between generalized Schur algebras, unifying classical and quantum cases, and applies these to category equivalences, structure analysis, and combinatorial identities.

Contribution

It introduces a characteristic-free construction of isomorphisms between generalized Schur algebras applicable to both classical and quantum cases, with broad algebraic and combinatorial applications.

Findings

Established characteristic-free isomorphisms between generalized Schur algebras.

Derived new results on decomposition numbers and $p$-Kostka numbers.

Reproved and extended row and column removal rules.

Abstract

Isomorphisms are constructed between generalized Schur algebras in different degrees. The construction covers both the classical case (of general linear groups over infinite fields of arbitrary characteristic) and the quantized case (in type $A$, for any non-zero value of the quantum parameter $q$). The construction does not depend on the characteristic of the underlying field or the choice of $q \neq 0$. The proof combines a combinatorial construction with comodule structures and Ringel duality. Applications range from equivalences of categories to results on the structure and cohomology of Schur algebras to identities of decomposition numbers and also of $p$-Kostka numbers, in both cases reproving and generalizing row and column removal rules.