tailieunhanh - Báo cáo toán học: " On plethysm conjectures of Stanley and Foulkes: the 2 × n case"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: On plethysm conjectures of Stanley and Foulkes: the 2 × n case. | On plethysm conjectures of Stanley and Foulkes the 2 X n case Pavlo Pylyavskyy Department of Mathematics MIT Massachusetts USA pasha@ Submitted Jul 5 2004 Accepted Oct 7 2004 Published Oct 18 2004 Mathematics Subject Classifications 05E10 Abstract We prove Stanley s plethysm conjecture for the 2 X n case which composed with the work of Black and List provides another proof of Foulkes conjecture for the 2 X n case. We also show that the way Stanley formulated his conjecture it is false in general and suggest an alternative formulation. 1 Introduction Denote by V a finite-dimensional complex vector space and by SmV its m-th symmetric power. Foulkes in 4 conjectured that the GL V -module Sn SmV contains the GL V -module Sm SnV for n m. For m 2 3 and 4 the conjecture was proved see 7 3 1 . An extensive list of references can be found in 8 . In 2 Black and List showed that Foulkes conjecture follows from the following combinatorial statement. Denote Im n to be the set of dissections of 1 . mn into sets of cardinality m. Let s I In 1 Si and t I Im 1 Ti be elements of Imn and Inm respectively. Define matrix Mm n Mff1 by Mm n 1 if Si e TjI 1 for any 1 i n1 j m t s ly otherwise. Theorem Black List 89 . If the rank of Mm n is equal to In m for n m 1 then Foulkes conjecture holds for all pairs of integers n r such that 1 r m. Let A be a partition of N. A tableau is a filling of a Young diagram of shape A with numbers from 1 to N and let T to be the set of such tableaux. Define two tableaux to be -equivalent denoted h if they can be obtained one from the other by permuting THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2004 R8 1 Figure 1 A counterexample for Stanley s conjecture. elements in rows and permuting rows of equal length. Define a horizontal tableau to be an element of Hx Tx h. In other words rows of a horizontal tableau form a partition of the set 1 . . Similarly define -equivalence v and the set Vx Tx v of vertical tableaux of shape A. Consider a .