## Two boundary Hecke Algebras and the combinatorics of type $C$

Last updated: 27 January 2015

## Contents

1 Introduction
2 The two-boundary Hecke algebra
 2.1 The two-boundary braid group 2.2 The two-boundary Hecke algebra ${H}_{k}^{\text{ext}}$ 2.3 Weights of representations and intertwiners
3 Calibrated representations of ${H}_{k}^{\text{ext}}$
 3.1 Local regions 3.2 Classifying and constructing calibrated representations
4 Classification of irreducible representations of ${H}_{2}$
 4.1 Classification of central characters 4.2 Construction of the irreducible ${H}_{2}\text{-modules}$
5 Representations of ${ℬ}_{k}^{\text{ext}}$ in tensor space
 5.1 Quantum groups and $R\text{-matrices}$ 5.2 The ${ℬ}_{k}^{\text{ext}}\text{-modules}$ ${B}_{k}^{\lambda }$ 5.3 Some tensor products for $𝔤={𝔤𝔩}_{n}$ 5.4 Irreducible ${H}_{k}^{\text{ext}}\text{-modules}$ in $M\otimes N\otimes {V}^{\otimes k}$

## Notes and References

This is an excerpt from the paper Two boundary Hecke Algebras and the combinatorics of type $C$ Zajj Daugherty (Department of Mathematics, The City College of New York, NAC 8/133, Convent Ave at 138th Street, New York, NY 10031) and Arun Ram (Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia).