tailieunhanh - Báo cáo toán học: "q, t-Catalan numbers and generators for the radical ideal defining the diagonal locus of (C2)n"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: q, t-Catalan numbers and generators for the radical ideal defining the diagonal locus of (C2)n. | q t-Catalan numbers and generators for the radical ideal defining the diagonal locus of C2 n Kyungyong Lee Department of Mathematics University of Connecticut Storrs CT 06269 . Li Li Department of Mathematics and Statistics Oakland University Rochester MI 48309 . li2345@ Submitted Dec 6 2010 Accepted Jul 28 2011 Published Aug 5 2011 Mathematics Subject Classifications 05E15 05E40 Abstract Let I be the ideal generated by alternating polynomials in two sets of n variables. Haiman proved that the q t-Catalan number is the Hilbert series of the bi-graded vector space M 0d1 d2 Md1 d2 spanned by a minimal set of generators for I. In this paper we give simple upper bounds on dim Md1 d2 in terms of number of partitions and find all bi-degrees d1 d2 such that dimMd1 d2 achieve the upper bounds. For such bi-degrees we also find explicit bases for Md1 d2. 1 Introduction In 6 Garsia and Haiman introduced the q t-Catalan number Cn q t and showed that Cn q 1 agrees with the q-Catalan number defined by Carlitz and Riordan 3 . To be more precise take the n X n square whose southwest corner is 0 0 and northeast corner is n n . Let Dn be the collection of Dyck paths . lattice paths from 0 0 to n n that proceed by NORTH or EAST steps and never go below the diagonal. For any Dyck path n define area n to be the number of lattice squares below n and strictly above the diagonal. Then C q 1 V qarea n . The q t-Catalan number Cn q t also has a combinatorial interpretation using Dyck paths. Given a Dyck path n let ai n be the number of squares in the i-th row that lie in the region bounded by n and the diagonal and define dinv n i j i j and ữựn aj n I i j i j and ữựn 1 aj n I. Partially supported by NSF grant DMS 0901367 THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P158 1 In 4 1 and 5 Theorem Garsia and Haglund showed the following combinatorial formula 1 Cn q t 2 qarea n tdinv n . A natural question is to find the coefficient of .