Divided differences for symmetric functions and alternating higher Bruhat orders (Q2784562)

The elements of higher Bruhat orders [\textit{Yu. I. Manin} and \textit{V. V. Shekhtman}, Adv. Stud. Pure Math. 17, 289-308 (1989; Zbl 0759.20002)] codify the connected components in the space of nondegenerate configurations of points in \(\mathbb{R}^2\) [see \textit{G. G. Il'yuta}, Izv. Math. 60, 1183-1192 (1996); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 60, 91-100 (1996; Zbl 0892.58007)]. The discriminantal hypersurface is determined in terms of Lagrange interpolation. The Lagrange interpolation formula is generalized to the case of symmetric functions in [\textit{W. Y. C. Chen} and \textit{J. D. Louck}, Adv. Math. 117, 147-156 (1996; Zbl 0857.05092)].NEWLINENEWLINENEWLINEThe purpose of the paper under review is to announce, without proofs, a generalized formula for divided differences for symmetric functions, expressing the divided difference in terms of the coefficients of the Lagrange interpolation formula. This suggests the introduction of generalized higher Bruhat orders which the author calls alternating because of the close relation with alternating oriented matroids. The theory of oriented matroids can be regarded as a combinatorial theory of determinantal identities and, from this point of view, the formula in the paper can be described by a formula for the product of maximal minors in a rectangular matrix. The author states that in a similar way one can generalize other formulas, for example, the Newton interpolation formula and the formula for divided differences of products and compositions of functions.