tailieunhanh - Báo cáo toán học: "A Relationship Between the Major Index For Tableaux and the Charge Statistic For Permutations"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A Relationship Between the Major Index For Tableaux and the Charge Statistic For Permutations. | A Relationship Between the Major Index For Tableaux and the Charge Statistic For Permutations Kendra Killpatrick Pepperdine University Malibu California Submitted Jul 13 2005 Accepted Aug 30 2005 Published Sep 5 2005 Mathematics Subject Classifications 05A15 05E10 Abstract The widely studied -polynomial f x q which specializes when q 1 to fx the number of standard Young tableaux of shape A has multiple combinatorial interpretations. It represents the dimension of the unipotent representation Sq of the finite general linear group GLn q it occurs as a special case of the Kostka-Foulkes polynomials and it gives the generating function for the major index statistic on standard Young tableaux. Similarly the q-polynomial gx q has combinatorial interpretations as the q-multinomial coefficient as the dimension of the permutation representation Mx of the general linear group GLn q and as the generating function for both the inversion statistic and the charge statistic on permutations in Wq. It is a well known result that for A a partition of n dim MqX Y K dim Sg where the sum is over all partitions g of n and where the Kostka number K x gives the number of semistandard Young tableaux of shape g and content A. Thus gx q fx q is a q-polynomial with nonnegative coefficients. This paper gives a combinatorial proof of this result by defining an injection f from the set of standard Young tableaux of shape A SYT A to Wx such that maj T ch f T for T 2 SYT A . Key words Young tableaux permutation statistics inversion statistic charge statistic Kostka polynomials. 1 Introduction For A any partition of n fx gives the number of standard Young tableaux of shape A. The q-version of fx is a polynomial that has many important combinatorial interpretations. In particular f x q is known to give the dimension of the unipotent representation Sq THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R45 1 of the finite general linear group GLn q . The polynomial f x q