Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Plethysm for wreath products and homology of sub-posets of Dowling lattices. | Plethysm for wreath products and homology of sub-posets of Dowling lattices Anthony Henderson School of Mathematics and Statistics University of Sydney NSW 2006 AUSTRALIA anthonyh@ Submitted Apr 5 2006 Accepted Oct 6 2006 Published Oct 12 2006 Mathematics Subject Classification 05E25 Abstract We prove analogues for sub-posets of the Dowling lattices of the results of Calder-bank Hanlon and Robinson on homology of sub-posets of the partition lattices. The technical tool used is the wreath product analogue of the tensor species of Joyal. Introduction For any positive integer n and finite group G the Dowling lattice Qn G is a poset with an action of the wreath product group G o Sn. If G is trivial Qn 1 can be identified with the partition lattice nn 1 on which Sn acts as a subgroup of Sn 1 . If G is the cyclic group of order r for r 2 Qn G can be identified with the lattice of intersections of reflecting hyperplanes in the reflection representation of G o Sn. For general G the underlying set of Qn G can be thought of as the set of all pairs I where I c 1 n and is a set partition of G X 1 n n I whose parts G permutes freely see Definition below for the partial order. In Section 1 we will define various sub-posets P of Qn G containing the minimum element 0 and the maximum element 1 which are stable under the action of G o Sn. For completeness sake we include the cases of Qn G itself and two other sub-posets which have been studied before but the main interest lies in two new families of sub-posets defined using a fixed integer d 2 Qnmodd G given by the congruence conditions II 0 mod d and KI 1 mod d for all parts K of and Qnmodd G given by the condition KI 0 mod d for all parts K of . These definitions are modelled on those of the sub-posets I l 1 d and nn d of the partition lattice studied by Calderbank Hanlon and Robinson in 4 . We will prove that all our sub-posets P are pure . graded and This work was supported by Australian Research Council
