Antipodes and incidence coalgebras
type: Item
instance of: scholarly article
title: Antipodes and incidence coalgebras
zbMATH Open document ID: 0699.05003
DOI: 10.1016/0097-3165(87)90006-9
author: William R. Schmitt
published in: Journal of Combinatorial Theory. Series A
publication date: 1987-01-01T00:00:00Z
review text: Let \({\mathcal P}\) be a collection of locally finite partially ordered sets. An equivalence relation \(\sim\) on the collection of all segments of intervals in \({\mathcal P}\) is said to be order-compatible if [x,y]\(\sim [u,v]\) implies that there exists a bijection \(\phi: [x,y]\to [u,v]\) such that for all z in \([x,y],[x,z]\sim [\phi(x),\phi(z)]\) and \([z,y]\sim [\phi(z),\phi(y)].\) The (reduced) incidence coalgebra \(C({\mathcal P}_{\sim})\) modulo \(\sim\) is the free vector space spanned by the variables \(X^{\alpha}\), where \(\alpha\) is a type or equivalence class with comultiplication given by \(\Delta X^{\alpha}= \sum_{\sigma} X^{\sigma^-}\otimes X^{\sigma^+}\), where the sum on the right is computed by taking a representative [x,y] in \(\alpha\) and summing over all z, \(x\leq z\leq y\), with \(\sigma^-\) the type of [x,z] and \(\sigma^+\) the type of [z,y] [cf. \textit{J. R. Goldman} and \textit{G.-C. Rota}, Stud. Appl. Math. 49, 239-258 (1970; Zbl 0212.033)]. Among the results are: (a) An equivalence relation (containing all isomorphisms) is order-compatible if and only if, in the incidence algebra modulo isomorphism, the subspace spanned by \(X^ P-X^ Q\), where \(P\sim Q\), is a coideal; (b) construction of the free incidence bialgebra; (c) conditions for an incidence coalgebra to be embeddable combinatorially into an incidence Hopf algebra; (d) a recursion and a closed formula for the antipode of an incidence Hopf algebra. For enumeration over posets, the most important result is probably the expression for the antipode as a sum over alternating chains. As the author observes, since the antipode of H is the two-sided inverse of the identity map \(H\to H\) under convolution, the antipode is an analogue of the Möbius function [see \textit{G.-C. Rota}, Z. Wahrsch. Verw. Gebiete 2, 340-368 (1964)] and the closed formula is reminiscent of \textit{P. Hall}'s formula for the Möbius function [Q. J. Math., Oxf. II. Ser. 7, 134-151 (1936; Zbl 0014.10402)]…
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 4145902
zbMATH Keywords: ordinary formal power series
zbMATH Keywords: inversion of Dirichlet series
zbMATH Keywords: locally finite partially ordered sets
zbMATH Keywords: Hopf algebra
zbMATH Keywords: Möbius function
zbMATH Keywords: bialgebras
zbMATH Keywords: inversion formula
MaRDI profile type: Publication
full work available at URL: 0097-3165(87)90006-9
OpenAlex ID: W2050134666
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description: scientific article; zbMATH DE number 4145902