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 2 -parameter type A A family that we call N C A ( n , d ) NC_A(n,d) . We explore several combinatorial properties of N C A ( n , d ) NC_A(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 N C A ( n , d ) 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 Broué–Malle–Rouquier (BMR) freeness conjecture [J. Reine Angew. Math. 1998], we define generalized nil-Coxeter algebras N C W NC_W over all discrete real or complex reflection groups W 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 N C A ( n , d ) 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 N C W NC_W for W W complex.