Combinatorial Representation Theory

This paper is primarily intended as an introduction for the mathematically inclined to some of the rich algebraic combinatorics arising in for instance CFT. It is essentially self-contained, apart from some of the background motivation and examples which are included to give the reader a sense of the context. The theory is still a work-in-progress, and emphasis is given here to several open questions and problems.

We give a survey of recent developments in the investigation of the various local-global conjectures for representations of finite groups.

In this survey article we summarize the current state of research in representation stability theory. We look at three different, yet related, approaches, using (1) the category of FI-modules, (2) Schur-Weyl duality, and (3) finitely-generated modules over certain infinite dimensional vector spaces. The main example is the stability of representations of the symmetric group, though there have also been some notable generalizations of representation stability to other groups. ...

This article has been replaced by arXiv:0807.3093

These are the notes of some lectures given by the author for a workshop held at TIFR, Mumbai in December, 2011, giving an exposition of the Deligne-Lusztig theory.

This is a companion paper to arXiv:0909.2280. It is mostly expository and focuses on the representation-theoretic and combinatorial aspects of the main problems considered in the other article.

The representation theory of the symmetric groups S_n is intimately related to combinatorics: combinatorial objects such as Young tableaux and combinatorial algorithms such as Murnaghan-Nakayama rule. In the limit as n tends to infinity, the structure of these combinatorial objects and algorithms becomes complicated and it is hard to extract from them some meaningful answers to asymptotic questions. In order to overcome these difficulties, a kind of dual combinatorics of the ...

In this paper we study the Frobenius characters of the invariant subspaces of the tensor powers of a representation V. The main result is a formula for these characters for a polynomial functor of V involving the characters for V. The main application is to representations V for which these characters are known. The best understood case is for V the vector representation of a symplectic group or special linear group. Other cases where there are some related results are the de...

This lecture note is intended to be a brief introduction to a recent development on the interplay between the ultradiscrete (or tropical) soliton systems and the combinatorial representation theory. We will concentrate on the simplest cases which admit elementary explanations without losing essential ideas of the theory. In particular we give definitions for the main constructions corresponding to the vector representation of type $A^{(1)}_1$.

This dissertation addresses several current problems in Representation Theory using crystal bases. It incorporates the results of arXiv:math.QA/0408113 and arXiv:math.RT/0603547, as well as previously unpublished results.