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 NCA(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 NCA(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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