Generalized Nil-coxeter Algebras over Discrete Complex Reflection Groups

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 NCA(n, d). We explore several combinatorial properties of NCA(n, d), including its Coxeter word basis, length function, and Hilbert–Poincaré series, and show that the corresponding generalized Coxeter group is not a flat deformation of NCA(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 Broué–Malle–Rouquier (BMR) freeness conjecture [J. reine angew. math. 1998], we define generalized nil-Coxeter algebras NCW over all discrete 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 NCA(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 NCW for W complex.