Splitting algebras and Schubert calculus (Q2841856)

The paper under review is concerned with a beautiful elegant algebraic framework to deal with the \textsl{intersection theory of flag schemes}, based on the theory of \textsl{universal factorization and/or splitting algebras} of a set of polynomials with coefficients in some unital commutative ring. Intersection theory of flag schemes is the key ingredient to prove many formulas which are useful to compute fundamental classes of degeneracy loci of maps of vector bundles (e.g. Thom--Porteous' formula or its generalization à la Kempf-Laksov).NEWLINENEWLINEThe exceptionally well written introduction illustrates aims and scopes of the paper and, at the same time, already provides a comprehensive description of the main results of Section 4, where the authors \textsl{``give the complete connection between the bivariant Chow group and Chow rings for flag schemes and Grassmannians on the one side, and splitting and factorization algebras on the other'')}.NEWLINENEWLINESection 1 is devoted to recall a few basic notions concerning \textsl{splitting} and \textsl{factorization} algebras. The notion of partial flag schemes is fully worked out in Section 2, whose intersection theory is treated in Section 3, exploiting the notion of bivariant Chow ring. The generalized Schur determinant of a locally free sheaf with respect to a partial flag is the key formula of this section. Chow groups for flag schemes are studied in Section 4 and entirely described in terms of suitable splitting algebras, those associated, roughly speaking, to the Chern polynomial of a locally free sheaf on some regular scheme.NEWLINENEWLINECorollary 4.5 turns out to be a transparent translation of the celebrated determinantal formula by Kempf and Laksov, which is stated again in Section 6 under the shape of a general Giambelli's-like formula. Another crucial notion emphasized by the authors is that of Gysin homomorphism which is related with the \textsl{divided differences operators}, studied in that famous paper by \textit{A. Lascoux} and \textit{M.-P. Schuetzenberger} [C. R. Acad. Sci., Paris, Sér. I 294, 447--450 (1982; Zbl 0495.14031)] where the double Schubert polynomials were introduced for the first time.NEWLINENEWLINEThe appendix sheds additional light on many combinatorial properties of symmetric functions, like the nice comparison with MacDonald's book on symmetric functions, Example 20, p. 54--57, which are useful for the proof of the important polarity formula stated in section 1.20. The paper concludes itself with a rich reference list which includes all the previous works the authors did in relation with the subject. Indeed, the origin of this paper is the main result appeared in [\textit{D. Laksov} and \textit{A. Thorup}, Indiana Univ. Math. J. 56, No. 2, 825--845 (2007; Zbl 1121.14045)], which proves that the canonical symmetric structure of the tensor power of a polynomial ring gets rid of the formalism of Schubert calculus for Grassmann bundles.NEWLINENEWLINEThe reviewer shares with the authors the opinion that the results proposed in this beautiful paper are definitely interesting for researchers in Algebra, Algebraic Geometry, Combinatorics and Representation Theory.