Hecke algebras with independent parameters

Abstract

Abstract. We study the Hecke algebra H(q) over an arbitrary field F of a Coxeter system (W, S ) with independent parameters q = (q s ∈ F : s ∈ S ) for all generators. This algebra is always linearly spanned by elements indexed by the Coxeter group W. This spanning set is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receive the same parameter. In general, the dimension of H(q) could be as small as 1. We construct a basis for H(q) when (W, S ) is simply laced. We also characterize when H(q) is commutative, which happens only if the Coxeter diagram of (W, S ) is simply laced and bipartite. In particular, for type A we obtain a tower of semisimple commutative algebras whose dimensions are the Fibonacci numbers. We show that the representation theory of these algebras has some features in analogy/connection with the representation theory of the symmetric groups and the 0-Hecke algebras.

A necessary and sufficient criterion for the existence of ratio limits of sequences generated by linear recurrences

“…Sloane's Online Encyclopedia of Integer Sequences [22] and Khovanova's website [16] catalog thousands of integer sequences generated by linear recurrences that are associated with problems in various branches of mathematics and other sciences, such as number theory, abstract algebra, linear algebra, combinatorics, complex numbers, group theory, probability, statistics, affine geometry, electrical networks, infectious diseases, etc., cf. [1,2,3,4,5,6,7,8,10,11,12,13,17,18,20,21,23,24,25].…”