Combinatorial Representation Theory

Combinatorial Representation Theory

We survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. Answers to the main questions are given in Part I for the fundamental structures, Sn and GL(n, C ), and later for certain generalizations, when known. Background material and more specialized results are given in a series of appendices. We give a per...

Nathaniel Thiem Combinatorial representation theory

My primary research interest is in the interplay between combinatorics and algebraic structures. By employing combinatorial tools such as symmetric functions, partitions, tableaux, graphs, posets, and crystal bases, one can gain significant insight on algebraic and geometric structures such as groups, algebras and rings; and, conversely, the corresponding structure theory can often lead to surp...

A New Combinatorial Model in Representation Theory

The present paper is a survey of a simple combinatorial model for the irreducible characters of complex semisimple Lie algebras, and, more generally, of complex symmetrizable Kac-Moody algebras. This model, which will be referred to as the alcove path model, can be viewed as a discrete counterpart to the Littelmann path model. It allows us to give character formulas and a LittlewoodRichardson r...

A Combinatorial Introduction to the Representation Theory of S n

Conditioned One-way Simple Random Walk and Combinatorial Representation Theory

A one-way simple random walk is a random walk in the quadrant Z+ whose increments are elements of the canonical base. In relation with representation theory of Lie algebras and superalgebras, we describe the law of such a random walk conditioned to stay in a closed octant, a semi-open octant, or other types of semi-groups. The combinatorial representation theory of these algebras allows us to d...