Schur indices and representation theory

代数在物理、编码理论和密码学等领域的许多现代应用依赖于对我们称为群、环和代数的代数结构的详细了解。在许多这些应用中，代数结构在它们的潜在应用实现之前就被很好地理解了。“舒尔指数和表示理论”是一个好奇心驱动的研究计划，旨在增加我们对如何将群和代数表示为具有非代数闭域(如有理数域)中的条目的矩阵的理解。Schur指数是一个不变量，它是详细理解这些表示的关键。程序的目标之一是开发一个计算机软件包，使有限群表示的Schur指数计算的困难过程对数学研究界来说是容易的。该程序的理论研究目标包括研究Schur指数领域的一种新的理论工具，称为G-代数的等价性，并研究关于Coxeter群表示的可能的Schur指数的猜想，该猜想在表示理论和离散几何中是重要的。

Many of the modern applications of algebra in areas including physics, coding theory, and cryptology depend on a detailed understanding of algebraic structures we call groups, rings, and algebras.  In many of these applications, the algebraic structures were understood well before their potential for applications was realized.  "Schur Indices and Representation Theory" is a curiousity-driven research program that seeks toincrease our understanding of how groups and algebras can be represented as matrices with entries in non-algebraically closed fields such as the field of rational numbers. The Schur index is the invariant which holds the key to a detailed understanding of these representations.  One of the goals of the program is the development of a computer software package that would make the difficult process of Schur index calculation for representations of finite groups accessible to the mathematical research community.  The theoretical research goals for this program include the study of a new theoretical tool in the area of Schur indices known as "equivalence of G-algebras", and investigations into a conjecture concerning the possible Schur indices for representations of Coxeter groups that are important in representation theory and discrete geometry.