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Séminaire Lotharingien de Combinatoire, 80B.29 (2018), 12 pp.

# Apoorva
Khare

# Generalized Nil-Coxeter Algebras

**Abstract.**
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
Brou\'e--Malle--Rouquier freeness conjecture (1998). With generic Hecke
algebras over real and complex groups in mind, we show that the first
`non-usual' 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.

Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.

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