Shifted Quasi-Symmetric Functions and the Hopf algebra of peak functions
Abstract
In his work on P-partitions, Stembridge defined the algebra of peak functions Pi, which is both a subalgebra and a retraction of the algebra of quasi-symmetric functions. We show that Pi is closed under coproduct, and therefore a Hopf algebra, and describe the kernel of the retraction. Billey and Haiman, in their work on Schubert polynomials, also defined a new class of quasi-symmetric functions --- shifted quasi-symmetric functions --- and we show that Pi is strictly contained in the linear span Xi of shifted quasi-symmetric functions. We show that Xi is a coalgebra, and compute the rank of the n-th graded component.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- April 1999
- DOI:
- 10.48550/arXiv.math/9904105
- arXiv:
- arXiv:math/9904105
- Bibcode:
- 1999math......4105B
- Keywords:
-
- Mathematics - Combinatorics;
- Mathematics - Rings and Algebras;
- 05E05;
- 16W30;
- 05E15;
- 05A15
- E-Print:
- 9 pages, 4 eps figures, uses epsf.sty. to be presented at FPSAC99 in Barcelona by second author