Combinatorial Representation Theory

@article{Barcelo1997CombinatorialRT,
  title={Combinatorial Representation Theory},
  author={H{\'e}l{\`e}ne Barcelo and Arun Ram},
  journal={Oberwolfach Reports},
  year={1997},
  volume={7},
  pages={799-882}
}
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. 

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References

SHOWING 1-10 OF 210 REFERENCES

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
...