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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Generalized nil-Coxeter algebras over discrete complex reflection groups
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by Apoorva Khare PDF
Trans. Amer. Math. Soc. 370 (2018), 2971-2999 Request permission


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:
  • 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:
  • MathSciNet review: 3748591