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Báo cáo toán học: "Bitableaux Bases for some Garsia-Haiman Modules and Other Related Modules"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Bitableaux Bases for some Garsia-Haiman Modules and Other Related Modules | Bitableaux Bases for some Garsia-Haiman Modules and Other Related Modules E. E. Allen Department of Mathematics Wake Forest University Winston-Salem NC 27109 allene@wfu.edu Submitted October 6 2000 Accepted April 5 2002. MR Subject Classifications 05E05 05E10 Abstract For certain subsets S and T of A 0 2 0 1 0 0 1 0 2 0 and factor spaces c S a Y c s T A Y Z W and c St A Y Z W bitableaux bases are constructed that are indexed by pairs of standard tableaux and sequences in the collections YộS and XộT. These bases give combinatorial interpretations to the appropriate Hilbert series of these spaces as well as the graded character of Cs AVY . The factor space cS A Y is an analogue of the coinvariant ring of a polynomial ring in two sets of variables. cS T A Y Z W an cS T A Y Z W are analogues of coinvariant spaces in symmetric and skew-symmetric polynomial settings respectively. The elements of the bitableaux bases are appropriately defined images in the polynomial spaces of bipermanents. The combinatorial interpretations of the respective Hilbert series and graded characters are given by statistics based on cocharge tableaux. Additionally it is shown that the Hilbert series and graded characters factor nicely. One of these factors gives the Hilbert series of a collection of Schur functions Sx p where ụ varies in an appropriately defined A. Thanks to all of the wonderful editors of this journal THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2002 R36 1 1. Introduction. Let A be the alphabet A J g f and g are nonnegative integers such that fg o . 1-1 Specihcally A o 3 o 2 o 1 o o 1 o 2 o 3 o . The elements of A are the coordinates i k of the cells of the hookshape in the hrst quadrant of the plane as shown 3 o 2 o 1 o o o o 1 o 2 o 3 o 4 o 5 We will say that if and only if ai bi a2 b2. Let C X Y Z W denote the polynomial ring with complex coefficients in X xi X2 Xn Y yi y2 yn Z zi Z2 Zn and W wi W2 Wn . Given a subset n o S ai bi a2 b2 an bnH G A of the alphabet A listed in .