Noncommutative symmetric functions of type B

The noncommutative symmetric functions Sym of Gelfand et al. give not only a lifting of the well-developed commutative theory of symmetric functions to the non-commutative level, but also relate the descent algebras of Solomon and the quasi-symmetric functions, where the latter are dual to the noncommutative symmetric functions equipped with the internal product, which are anti-isomorphic to the descent algebras. Using this anti-isomorphism, properties of both noncommutative symmetric functions and of descent algebras can be studied. Generalizations of the above theory are made in the present work. The starting point is the quasi-symmetric functions of type B, BQSym, which are shown to have an algebra, a comodule, and a coalgebra structures. The noncommutative symmetric functions BSym are then introduced as a module over Sym dual to the comodule structure of BQSym. It is then made into a coalgebra dual to the algebra structure of BQSym, and into an algebra dual to the coalgebra structure of BQSym. The latter duality defines the internal product *B on BSym, which makes (BSym, *B) anti-isomorphic to the descent algebra [Sigma]Bn of the hyperoctahedral groups Bn, studied by Bergeron and Bergeron.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2001

Author dc:creator

• Chow, Chak-On, 1968-

Advisor dc:contributor.advisor

• Ira M. Gessel.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/8640

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/8640