The lattice of nil-Hecke algebras over real and complex reflection groups

Associated to every complex reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, and which are obtained by killing all braid words that are "sufficiently long", as well as some integer power of each generator. These include usual nil-Coxeter algebras, nil-Temperley-Lieb algebras, and their variants, and lead to symmetric semigroup module categories which necessarily cannot be monoidal. Motivated by classical work of Coxeter (1957) and the Broue-Malle-Rouquier freeness conjecture [Crelle 1998], and continuing beyond work of the second author [Trans. Amer. Math. Soc. 2018], we obtain a complete classification of the finite-dimensional nil-Hecke algebras for all complex reflection groups $W$. These comprise the usual nil-Coxeter algebras for $W$ of finite type, their "fully commutative" analogues for $W$ of FC-finite type, three exceptional algebras (of types $F_4,H_3,H_4$), and three exceptional series (of types $B_n$ and $A_n$, two of them novel). In particular, we find the first - and only two - finite-dimensional nil-Hecke algebras over discrete complex reflection groups; this breaks from the nil-Coxeter case (where no braid words are further killed, and) where Marin [J. Pure Appl. Alg. 2014] and Khare [Trans. Amer. Math. Soc. 2018] showed that such algebras do not exist. In addition to these algebras, and also algebraic connections (to PBW deformations and non-monoidal tensor categories), we further uncover combinatorial bases of algebras, both known (fully commutative elements) and novel ($\bar{12}$-avoiding signed permutations). Our classification draws from and brings together results of Popov [Comm. Math. Inst. Utrecht 1982], Stembridge [J. Alg. Combin. 1996, 1998], Malle [Transform. Groups} 1996], Postnikov via Gowravaram-Khovanova (2015), Hart [J. Group Th. 2017], and Khare [Trans. Amer. Math. Soc. 2018].