Combinatorial Representation Theory

  title={Combinatorial Representation Theory},
  author={H{\'e}l{\`e}ne Barcelo and Arun Ram},
  journal={Oberwolfach Reports},
We attempt to survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. We give a personal viewpoint on the field, while remaining aware that there is much important and beautiful work that we have not been able to mention. 

Figures from this paper

The many faces of modern combinatorics

This is a survey of recent developments in combinatorics. The goal is to give a big picture of its many interactions with other areas of mathematics, such as: group theory, representation theory,

Modular representations, old and new

The modular representation theory of finite groups has its origins in the work of Richard Brauer. In this survey article, we first discuss the work being done on some outstanding conjectures in the

Posets , Weyl Characters , and Representations of Semisimple Lie Algebras

This work-in-progress is intended as an exposition of the background material, results, and open problems of a particular poset theoretic study of Weyl characters and semisimple Lie algebra

Polynomial Representations of the Hecke Algebra of the Symmetric Group

We give a polynomial basis of each irreducible representation of the Hecke algebra, as well as an adjoint basis. Decompositions in these bases are obtained by mere specializations.

A Survey of Representation Stability Theory

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)

Cellular subalgebras of the partition algebra

. We describe various diagram algebras and their representation theory using cellular algebras of Graham and Lehrer and the decomposition into half diagrams. In particular, we show the diagram

Twisted representations for -local vertex operator algebras

Abstract We introduce the notion of twisted module for -local vertex operator algebra and give a complete set of inequivalent irreducible admissible -twisted -modules when g is in the center of G,

Fusion procedure for the Brauer algebra

We show that all primitive idempotents for the Brauer algebra B_n(w) can be found by evaluating a rational function in several variables which has the form of a product of R-matrix type factors. This

s-Catalan numbers and Littlewood-Richardson polynomials

  • William Linz
  • Mathematics
    Enumerative Combinatorics and Applications
  • 2021
: In this note, we study two generalizations of the Catalan numbers, namely the s -Catalan numbers and the spin s -Catalan numbers. These numbers first appeared in relation to quantum physics problems



Representation Theory: A First Course

This volume represents a series of lectures which aims to introduce the beginner to the finite dimensional representations of Lie groups and Lie algebras. Following an introduction to representation

Combinatorial aspects of skew representations of the symmetric group

Characters of reductive groups over a finite field

This book presents a classification of all (complex) irreducible representations of a reductive group with connected centre, over a finite field. To achieve this, the author uses etale intersection

Representations of reductive groups over finite fields

In this paper we construct a family of irreducible representations of a Chevalley group over a finite ring R of truncated power series over a field Fq . This is done by a cohomological method

Real reductive groups

The Cauchy identity for Sp(2n)

  • S. Sundaram
  • Mathematics, Computer Science
    J. Comb. Theory, Ser. A
  • 1990

A schensted-type correspondence for the symplectic group

  • A. Berele
  • Mathematics
    J. Comb. Theory, Ser. A
  • 1986

An algorithm for characters of Hecke algebras Hn(q) of type An-1

Recently the characters of irreducible representations of the Hecke algebra Hn(q) of type An-1 were identified with the transition coefficients relating q-generalized power sum symmetric functions to

Methods of representation theory--with applications to finite groups and orders

Algebraic K -Theory. Class Groups of Integral Group Rings and Orders. The Theory of Blocks. The Representation Theory of Finite Groups of Lie Type. Rationality Questions. Indecomposable Modules. The