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Báo cáo toán học: "Grothendieck bialgebras, Partition lattices, and symmetric functions in noncommutative variables"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Grothendieck bialgebras, Partition lattices, and symmetric functions in noncommutative variables. | Grothendieck bialgebras Partition lattices and symmetric functions in noncommutative variables N. Bergeron 1 C. Hohlweg 2 M. Rosas 1 and M. Zabrocki 1. Department of Mathematics and Statistics York University Toronto Ontario M3J 1P3 Canada. bergeron@mathstat.yorku.ca mrosas@us.es zabrocki@mathstat.yorku.ca 2The Fields Institute 222 College Street Toronto Ontario M5T 3J1 Canada. chohlweg@fields.utoronto.ca Submitted Jul 14 2005 Accepted Jul 19 2006 Published Aug 25 2006 Mathematics Subject Classifications 05E05 05E10 16G10 20C08. Abstract We show that the Grothendieck bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of symmetric functions in noncommutative variables. In particular this isomorphism singles out a canonical new basis of the symmetric functions in noncommutative variables which would be an analogue of the Schur function basis for this bialgebra. Introduction Combinatorial Hopf algebras are graded connected Hopf algebras equipped with a multiplicative linear functional H k called a character see 1 . Here we assume that k is a field of characteristic zero. There has been renewed interest in these spaces in recent papers see for example 3 4 6 11 13 and the references therein . One particularly interesting aspect of recent work has been to realize a given combinatorial Hopf algebra as the Grothendieck Hopf algebra of a tower of algebras. The prototypical example is the Hopf algebra of symmetric functions viewed via the Frobenius characteristic map as the Grothendieck Hopf algebras of the modules of all This work is supported in part by CRC and NSERC. It is the results of a working seminar at Fields Institute with the active participation of T. MacHenry M. Mishna H. Li and L. Sabourin THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R75 1 symmetric group algebras kSb for n 0. The multiplication is given via induction from kSb G kSm to kSra m and the comultiplication is the sum over r of the .