Abstract
We are interested in the McKay quiver Γ(G) and skew group rings A ∗G, where G is a finite subgroup of GL(V ), where V is a finite dimensional vector space over a field K, and A is a K −G-algebra. These skew group rings appear in Auslander’s version of the McKay correspondence. In the first part of this paper we consider complex reflection groups \(\mathsf {G} \subseteq \text {GL}(V)\) and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups G(r,p,n). We first look at the case G(1,1,n), which is isomorphic to the symmetric group Sn, followed by G(r,1,n) for r > 1. Then, using Clifford theory, we can determine the McKay quiver for any G(r,p,n) and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra \(\widetilde {A}(\mathsf {G})\) of a finite group \(\mathsf {G} \subseteq \text {GL}(V)\), which is Morita equivalent to the skew group ring A ∗G. This description gives us an embedding of the basic algebra Morita equivalent to A ∗ G into a matrix algebra over A.
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Acknowledgements
The first three authors want to thank the Mathematisches Forschungszentrum Oberwolfach for the perfect working conditions and inspiring atmosphere. Most of the work for this paper was done in frame of the Leibniz fellowship programme and the Research in Pairs programme. The second author wants to thank Colin Ingalls and Carleton University for their hospitality. We also want to thank the anonymous referee for helpful comments. Last but not least, we thank Ruth Buchweitz for her hospitality and support.
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Ragnar-Olaf Buchweitz passed away on November 11, 2017.
The first and third authors were partially supported by an NSERC Discovery grant. The second author gratefully acknowledges support by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 789580.
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Buchweitz, RO., Faber, E., Ingalls, C. et al. McKay Quivers and Lusztig Algebras of Some Finite Groups. Algebr Represent Theor 26, 433–469 (2023). https://doi.org/10.1007/s10468-021-10099-x
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DOI: https://doi.org/10.1007/s10468-021-10099-x