GENERALIZED NIL-COXETER ALGEBRAS OVER DISCRETE COMPLEX REFLECTION GROUPS

Khare, Apoorva (2018) GENERALIZED NIL-COXETER ALGEBRAS OVER DISCRETE COMPLEX REFLECTION GROUPS. In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 370 (4). pp. 2971-2999.

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Abstract

We define and study generalized nil-Coxeter algebras associated to Coxeter groups. Motivated by a question of Coxeter (1957), we construct the first examples of such finite-dimensional algebras that are not the ``usual'' nil-Coxeter algebras: a novel 2-parameter type A family that we call NCA(n, d). We explore several combinatorial properties of NCA(n, d), including its Coxeter word basis, length function, and Hilbert-Poincare series, and show that the corresponding generalized Coxeter group is not a flat deformation of NCA(n, d). These algebras yield symmetric semigroup module categories that are necessarily not monoidal; we write down their Tannaka-Krein duality. Further motivated by the Broue-Malle-Rouquier (BMR) freeness conjecture J. Reine Angew. Math. 1998], we define generalized nil-Coxeter algebras NCW over all discrete real or complex reflection groups We, finite or infinite. We provide a complete classification of all such algebras that are finite dimensional. Remarkably, these turn out to be either the usual nil-Coxeter algebras or the algebras NCA(n, d). This proves as a special case-and strengthens-the lack of equidimensional nil-Coxeter analogues for finite complex reflection groups. In particular, generic Hecke algebras are not flat deformations of NCW for W complex.

Item Type:	Journal Article
Publication:	TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
Additional Information:	Copy right for this article belong to the AMER MATHEMATICAL SOC, 201 CHARLES ST, PROVIDENCE, RI 02940-2213 USA
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	02 Mar 2018 15:03
Last Modified:	03 Oct 2018 14:57
URI:	http://eprints.iisc.ac.in/id/eprint/58933

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