RUI - Combinatorial Methods in Representation Theory

Macalester College, Saint Paul MN

Investigators

Abstract

The goal of this project is to use combinatorial methods to solve problems in the field of matrix representations of groups and algebras, and, conversely, to use representation theory to produce new results in enumerative combinatorics. A main focus of each project is to construct and understand irreducible representations and characters. The algebraic structures under investigation include symmetric groups and Weyl groups, the finite general linear group and other finite Chevalley groups, partition algebras, and Hecke algebras. These structures often come as pairs acting on vector spaces and commuting with each other; for example the symmetric group and the partition algebra commute with each other on tensor space and a Chevalley group and its Hecke algebra commute on a flag variety. This duality allows us to bring known information from one of the algebraic structures to bear on the study of its partner. The combinatorial structures used here include symmetric functions, partitions, tableaux, integer matrices, and graphs. This project will engage undergraduate students in all aspects of the research. Algebraic combinatorics is an ideal way to introduce young mathematicians and scientists to research, because it uses accessible combinatorial objects, which can be created, analyzed, and manipulated by students who do not have of a large amount of advanced training. At the same time, these objects enhance the fundamental understanding of important constructs from algebra, geometry, and topology. The algebraic objects under investigation in this project represent various kinds of symmetry. These include symmetries of physical objects --- such as molecules, crystals, knots, or tensegrities --- and symmetries of complicated systems, such as physical models of energy transfer, large sets of ranked data, or symmetry in the organization of digital image data. Representation theory ``represents" these symmetries using matrices (rectangular arrays of numbers ). In a given problem, there are usually a large number of these matrices and they also are usually large in dimension (size). A major goal of combinatorial representation theory is to find ways to simultaneously decompose these large matrices into their constituent building blocks and then to analyze the building blocks using concrete structures which can be manipulated by hand and by computer. This is analogous to understanding a complicated molecule by decomposing it into the various atoms that make it up and then carefully analyzing those atoms. Since the early 20th century, representation theory has held a prominent role in theoretical mathematics. Furthermore, it has had an enormous impact on physical chemistry, particle physics, and knot theory. In recent years, the areas of application of representation theory have grown to include data analysis, digital image processing, coding theory, and engineering.