Antipodes and involutions

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• 作者：Carolina Benedetti ^{caro.benedetti@gmail.com} ; Bruce E. Sagan ^{sagan@math.msu.edu}
• 关键词：Acyclic orientation ; Antipode ; Involution ; Malvenuto&ndash ; Reutenauer Hopf algebra ; mQSymmQSym ; NSym ; Poirier&ndash ; Reutenauer Hopf algebra ; QSym ; Shuffle Hopf algebra ; Takeuchi formula
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：May 2017
• 年：2017
• 卷：148
• 期：Complete
• 页码：275-315
• 全文大小：630 K
• 卷排序：148

文摘

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 multi-quasisymmetric 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 Poirier–Reutenauer Hopf algebras, some of which are the first of their kind. We include various conjectures and suggestions for future research.