Abstract
Properties of relative traces and symmetrizing forms on chains of cyclotomic and affine Hecke algebras are studied. The study relies on a use of bases of these algebras which generalize a normal form for elements of the complex reflection groups $G(m,1,n)$, $m=1,2,\dots,\infty$, constructed by a recursive use of the Coxeter--Todd algorithm. Formulas for inducing, from representations of an algebra in the chain, representations of the next member of the chain are presented.
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