Generalized nil-Coxeter algebras over discrete complex reflection groups
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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 $NC_A(n,d)$. We explore several combinatorial properties of $NC_A(n,d)$, including its Coxeter word basis, length function, and Hilbert–Poincaré series, and show that the corresponding generalized Coxeter group is not a flat deformation of $NC_A(n,d)$. These algebras yield symmetric semigroup module categories that are necessarily not monoidal; we write down their Tannaka–Krein duality.
Further motivated by the Broué–Malle–Rouquier (BMR) freeness conjecture [J. Reine Angew. Math. 1998], we define generalized nil-Coxeter algebras $NC_W$ over all discrete real or complex reflection groups $W$, 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 $NC_A(n,d)$. 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 $NC_W$ for $W$ complex.
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Additional Information
- Apoorva Khare
- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore – 560012, India
- MR Author ID: 750359
- ORCID: 0000-0002-1577-9171
- Email: khare@iisc.ac.in
- Received by editor(s): April 6, 2017
- Received by editor(s) in revised form: May 30, 2017
- Published electronically: November 28, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 2971-2999
- MSC (2010): Primary 20F55; Secondary 20F05, 20C08
- DOI: https://doi.org/10.1090/tran/7304
- MathSciNet review: 3748591