Let \(G\) be a compact connected Lie group and \(H\) be the centralizer of a one-parameter subgroup in \(G\). The problem posed by the author in section 1 is to determine numbers \( a_{u,v}^w \) appearing in the classical cases such as the Littlewood-Richardson rule, the Chevalley formula and the classical Pieri formula in terms of certain Cartan numbers of \(G\). His main result is stated as a theorem in subsection 2.3 providing a formula to compute Schubert classes.
The rest of the paper is organized as follows. In subsection 2.4 the author introduces an algorithm to compute the numbers \(a_{u,v}^w\). In section 3 he develops preliminary results from algebraic topology in which he defines in subsection 3.3 a special class of spaces known as oriented twisted product of 2-spheres of rank \(k\). In section 4, 2-spherical involutions are studied (subsection 4.3) which allow to construct the so-called Bott-Samelson cycles associated to a sequence of \(k\) roots (definition 6 of section 4.4). In section I, the image of a Schubert class as an element of the cohomology ring of the associated cycle in lemma 5.1 is given and proved in section 5.1. In section 5.2 the author defines a Bott-Samelson resolution of \(X_w\) for \( w \in W \), the Weyl group of \(G\), completing the proof of lemma 5.1 at the end of subsection 5.4. Section 6 is devoted to the proof of the main theorem of the paper. For that he uses lemma 5.1 and the description of the additive map given in section 3.4 lemma 3.4 in terms of the operator \(T_A\) defined in subsection 2.2. Section 7 recalls developments of a more historical nature used in this paper. In subsection 7.1 the author states as proposition 2 the generalization of \(K\)-cycles of R. Bott, H. Samelson [Am. J. Math. 80, 964–1029 (1958; Zbl 0101.39702)] to Bott-Samelson cycles as here defined. In subsection 7.3 he recalls the Bruhat-Chevalley decomposition [C. Chevalley, in: Algebraic groups and their generalization, Proc. Symp. Pure Math. 56, 1–26 (1994; Zbl 0824.14042)] of the quotient \(K/B\) where \(K\) is a linear algebraic group and \(B \subset K \) is a Borel subgroup in terms of the open cells of Schubert varieties. In subsection 7.4, definition 7.4 he gives the version of the Schubert variety in \(G / T\) used in this paper. In subsection 7.5 a linkage between the operator introduced in definition 3 of subsection 3.2 and the divided difference operator given by I. N. Bernstein et al. [Russ. Math. Surv. 28, 1–26 (1973; Zbl 0289.57024)] and M. Demazure [Invent. Math. 21, 287–301 (1973; Zbl 0269.22010)] is given in proposition 6. The author concludes the paper in subsection 7.6 commenting that his formula falls short of positivity using the additive basis introduced in lemma 3.3.