2021/11/29 by Bhattacharya, Sutanay, Khare, Apoorva
#20C08 (Secondary) #20F05 #20F55 (Primary) #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2111.14800
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 F4,H3,H4), and three exceptional series (of types Bn and An, 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 (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].