Generalized nil-Coxeter algebras over discrete complex reflection groups

Abstract

We define and study generalized nil-Coxeter algebras associated to Coxeter groups. Motivated by a question of Coxeter (1957), we construct the first examples of such finite-dimensional algebras that are not the 'usual' nil-Coxeter algebras: a novel 2-parameter type $A$ family that we call $NC_A(n,d)$. We explore several combinatorial properties of $NC_A(n,d)$, including its Coxeter word basis, length function, and Hilbert-Poincare series, and show that the corresponding generalized Coxeter group is not a flat deformation of $NC_A(n,d)$. These algebras yield symmetric semigroup module categories that are necessarily not monoidal; we write down their Tannaka-Krein duality. Further motivated by the Broue-Malle-Rouquier (BMR) freeness conjecture [J. reine angew. math. 1998], we define generalized nil-Coxeter algebras over all discrete real or complex reflection groups $W$, finite or infinite. We provide a complete classification of all such algebras that are finite-dimensional. Remarkably, these turn out to be either the usual nil-Coxeter algebras, or the algebras $NC_A(n,d)$. This proves as a special case - and strengthens - the lack of equidimensional nil-Coxeter analogues for finite complex reflection groups. In particular, generic Hecke algebras are not flat deformations of $NC_W$ for $W$ complex.