Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A semigroup approach to wreath-product extensions of Solomon’s descent algebras. | A semigroup approach to wreath-product extensions of Solomon s descent algebras Samuel K. Hsiao Mathematics Program Bard College Annandale-on-Hudson NY 12504 hsiao@ Submitted Aug 15 2008 Accepted Jan 27 2009 Published Feb 4 2009 Mathematics Subject Classification 05E99 16S34 20M25 Abstract There is a well-known combinatorial model based on ordered set partitions of the semigroup of faces of the braid arrangement. We generalize this model to obtain a semigroup FG associated with G o Sn the wreath product of the symmetric group Sn with an arbitrary group G. Techniques of Bidigare and Brown are adapted to construct an anti-homomorphism from the Sn-invariant subalgebra of the semigroup algebra of Ff into the group algebra of G o Sn. The colored descent algebras of Mantaci and Reutenauer are obtained as homomorphic images when G is abelian. 1 Introduction A celebrated result of Solomon 27 reveals the existence of an intriguing subalgebra known as the descent algebra inside the group algebra of any finite Coxeter group. In the case of the symmetric group the descent algebra has a particularly simple combinatorial interpretation in terms of descent sets of permutations. This interpretation is an important ingredient in numerous extensions applications and further investigations 13 5 12 18 22 . A fitting example one that is central to this paper is Mantaci and Reutenauer s construction of colored descent algebras 18 via wreath-product extensions of the symmetric group. Their work highlights the vibrant interest in developing colored versions of combinatorial tools associated with the symmetric group. Along these lines a significant development is Baumann and Hohlweg s 4 far-reaching descent theory for wreath products in which the functorial nature of the colored constructions are brought to light. Continuing in this vain Bergeron and Hohlweg 6 provide a unifiying generalization of a number of colored constructions in the literature and discover new colored .
