Combinatorial aspects of Hecke algebra characters

Iwahori-Hecke algebras are deformations of Coxeter group algebras. Their origins lie in the theory of automorphic forms but they arise in the representation theory of Coxeter groups and Lie algebras and in quantum group theory. The Kazhdan-Lusztig bases of these algebras, originally introduced in the late 1970s in connection with representation-theoretic concerns, has turned out to have deep connections to Schubert varieties, intersection cohomology, and related topics.Matrix immanants were originally introduced by Littlewood as a generalization of determinants and permanants. They remained obscure until the 1980s when their connections to symmetric function and representation theory as well as their surprising algebraic and combinatorial properties came to light. In particular, it was discovered that they have a fruitful connection to the theory of total positivity. More recently, a theory of quantum immamants was developed, providing a bridge to the quantum group theory.In this paper we develop the theory of certain planar networks, which provide a unified combinatorial setting for these fields of study. In particular, we use these networks to evaluate certain characters of the symmetric group algebra. We give new combinatorial interpretations of the quantum induced sign and trivial characters of the type A Iwahori-Hecke algebras.