TY - JOUR

T1 - The quiver of an algebra associated to the Mantaci-Reutenauer descent algebra and the homology of regular semigroups

AU - Margolis, Stuart

AU - Steinberg, Benjamin

N1 - Funding Information:
The second author was supported in part by NSERC.

PY - 2011/2

Y1 - 2011/2

N2 - 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.

AB - 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.

KW - Descent algebras

KW - Quivers

KW - Regular semigroups

KW - Representation theory

UR - http://www.scopus.com/inward/record.url?scp=79951677410&partnerID=8YFLogxK

U2 - 10.1007/s10468-009-9181-2

DO - 10.1007/s10468-009-9181-2

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AN - SCOPUS:79951677410

SN - 1386-923X

VL - 14

SP - 131

EP - 159

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

IS - 1

ER -