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The Quiver of an Algebra Associated to the Mantaci-Reutenauer Descent Algebra and the Homology of Regular Semigroups

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Abstract

We develop the homology theory of the algebra of a regular semigroup, which is a particularly nice case of a quasi-hereditary algebra in good characteristic. Directedness is characterized for these algebras, generalizing the case of semisimple algebras studied by Munn and Ponizovksy. We then apply homological methods to compute (modulo group theory) the quiver of a right regular band of groups, generalizing Saliola’s results for a right regular band. Right regular bands of groups come up in the representation theory of wreath products with symmetric groups in much the same way that right regular bands appear in the representation theory of finite Coxeter groups via the Solomon-Tits algebra of its Coxeter complex. In particular, we compute the quiver of Hsiao’s algebra, which is related to the Mantaci-Reutenauer descent algebra.

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Authors and Affiliations

  1. Department of Mathematics, Bar Ilan University, 52900, Ramat Gan, Israel

    Stuart Margolis & Benjamin Steinberg

  2. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6, Canada

    Stuart Margolis & Benjamin Steinberg

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  1. Stuart Margolis
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  2. Benjamin Steinberg
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Correspondence to Stuart Margolis.

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Dedicated to the memory of W. D. Munn.

The second author was supported in part by NSERC.

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Margolis, S., Steinberg, B. The Quiver of an Algebra Associated to the Mantaci-Reutenauer Descent Algebra and the Homology of Regular Semigroups. Algebr Represent Theor 14, 131–159 (2011). https://doi.org/10.1007/s10468-009-9181-2

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  • DOI: https://doi.org/10.1007/s10468-009-9181-2

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