  Motivated by work of Coxeter (1957), we study a class of algebras associated
to Coxeter groups, which we term 'generalized nil-Coxeter algebras'. We
construct the first finite-dimensional examples other than usual nil-Coxeter
algebras; these form a $2$-parameter type $A$ family that we term $NC_A(n,d)$.
We explore the combinatorial properties of these algebras, including the
Coxeter word basis, length function, maximal words, and their connection to
Khovanov's categorification of the Weyl algebra.
  Our broader motivation arises from complex reflection groups and the
Broue-Malle-Rouquier freeness conjecture (1998). With generic Hecke algebras
over real and complex groups in mind, we show that the 'first'
finite-dimensional examples $NC_A(n,d)$ are in fact the only ones, outside of
the usual nil-Coxeter algebras. The proofs use a diagrammatic calculus akin to
crystal theory.
