Homology of Newtonian coalgebras.
The \({\mathbf c}{\mathbf d}\)-index of a convex polytope encodes information about its face-lattice in a non-redundant way. In [J. Algebr. Comb. 8, No. 3, 273--299 (1998; Zbl 0917.06001)], the authors observed that this is a coalgebra homomorphism, so that geometric operations on a polytope can be expressed as derivations on the \({\mathbf c}{\mathbf d}\)-index. Others, with these authors, have followed these ideas up. In the present paper, the authors work with two Newtonian coalgebras \(A\) and their Newtonian chain complexes \({\mathcal N}_n(A)\): first, \(R\langle{\mathbf a},{\mathbf b}\rangle\), generated by the noncommutative variables \({\mathbf a}\) and \({\mathbf b}\), and, second, \(R\langle{\mathbf c}, {\mathbf d}\rangle\), generated by \({\mathbf c}:= {\mathbf a}+ {\mathbf b}\) and \({\mathbf d}:= {\mathbf a}{\mathbf b}+ {\mathbf b}{\mathbf a}\). The homology of \(R\langle{\mathbf a}, {\mathbf b}\rangle\) corresponds to the reduced homology of the boundary complex of the cross-polytope, while that of \(R\langle{\mathbf c}, {\mathbf d}\rangle\) depends on whether or not \(2\) is a unit in the underlying ring \(R\). In the case \(R= \mathbb{Z}\), the authors obtain a new proof of the result of \textit{M. M. Bayer} and \textit{A. Klapper} [Discrete Comput. Geom. 6, No. 1, 33--47 (1991; Zbl 0761.52009)], which shows that the \({\mathbf c}{\mathbf d}\)-index of an Eulerian poset exists.
scientific article; zbMATH DE number 1884176
scientific article