Generalized nil-Coxeter algebras over discrete complex reflection groups

Khare, Apoorva

doi:10.1090/tran/7304

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

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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