Antipodes and involutions

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:148

Article

Benedetti, Carolina¹  Sagan, Bruce E.¹ 

[1] Michigan State Univ, Dept Math, E Lansing, MI 48824 USA

关键词: Acyclic orientation;    Antipode;    Involution;    Malvenuto-Reutenauer Hopf algebra;    mQSym;    NSym;    Poirier-Reutenauer Hopf algebra;    QSym;    Shuffle Hopf algebra;    Takeuchi formula;   

DOI  :  10.1016/j.jcta.2016.12.005

来源: Elsevier

【 摘 要 】

If H is a connected, graded Hopf algebra, then Takeuchi's formula can be used to compute its antipode. However, there is usually massive cancellation in the result. We show how sign-reversing involutions can sometimes be used to obtain cancellation-free formulas. We apply this idea to nine different examples. We rederive known formulas for the antipodes in the Hopf algebra of polynomials, the shuffle Hopf algebra, the Hopf algebra of quasisymmetric functions in both the monomial and fundamental bases, the Hopf algebra of multiquasisymmetric functions in the fundamental basis, and the incidence Hopf algebra of graphs. We also find cancellation free expressions for particular values of the antipode in the immaculate basis for the noncommutative symmetric functions as well as the Malvenuto-Reutenauer and PoirierReutenauer Hopf algebras, some of which are the first of their kind. We include various conjectures and suggestions for future research. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free