The paper under review is concerned with a subject of very much current interest, namely that regarding the determination of the constants structure of the equivariant cohomology of the Grassmannian variety \(G(k,n)\), parameterizing \(k\)-dimensional vector subspaces of \({\mathbb C}^n\). It is related with some previous work on the same subject done by \textit{A. Knutson} and \textit{T. Tao} [Duke Math. J. 119, 221--260 (2003; Zbl 1064.14063)] and \textit{L. C. Mihalcea} [Adv. Math. 203, 1--33 (2006; Zbl 1100.14045) and Trans. Am. Math. Soc. 360, No. 5, 2285--2301 (2008; Zbl 1136.14046)], but it takes a different, original and completely new point of view. To explain it, recall that the \(k\)-th tensor power \(\bigotimes^kA[X]\) is naturally a module over the ring \(S\) of symmetric functions in \(k\)-indeterminates. Then, in a previous joint paper with \textit{A.~Thorup} [Indiana Univ. Math. J. 56, No. 2, 825--845 (2007; Zbl 1121.14045)], the author proves that \(\bigwedge^kA[X]\), the \(k\)-th exterior power of a polynomial ring with coefficients in a commutative ring with unit, carries a symmetric structure, which is the unique \(S\)-module structure such that the canonical \(A\)-epimorphism \(\bigotimes^kA[X]\rightarrow \bigwedge^kA[X]\) is \(S\)-linear as well. It turns out that \(\bigwedge^kA[X]\) is a free \(S\)-module of rank \(1\) and such a module structure is isomorphic to the homology of the infinite Grassmannian seen as a module over the cohomology via cap product. The cohomology of finite Grassmannians is achieved by the symmetric structure on the \(k\)-th exterior power of the quotient ring \(A[X]/(X^{n})\). The most important result of the paper under review is that it supplies, using the same author's words, ``a formalism for equivariant cohomology where the basic results of equivariant Schubert calculus, the basis theorem, Pieri's formula and Giambelli's formula can be obtained from the corresponding results of the general framework by a change of basis.'' Indeed the author extends such a result to include Mihalcea's formulation of quantum equivariant Schubert calculus as well. The paper is organized as follows. It begins with a very explanatory introduction relating the work done with the, more or less, recent literature on the subject. Section 1 is devoted to recall the main features of the factorization and splitting algebras of a given polynomial. Section~2 already achieves what the author calls equivariant Schubert Calculus on \(\bigwedge^kA[X]\), where in particular he gets a Giambelli's equivariant formula involving the factorial Schur functions in its expression. The main point is that this formula is the same borrowed from the general Schubert calculus on an exterior paper developed in the aforementioned joint paper with Thorup. Such a formula offers the pretest for devoting section~3 to a closer view of the properties of the factorial Schur functions, while Section~4 relates them with the Schubert classes in the equivariant context, drawing a very detailed comparison with part of the work by Knutson and Tao. Equivariant Pieri's formulas, never published in any previous literature, are displayed in Section~5 while Section~6 builds the due bridge with geometry. The inclusion of (part of) Mihalcea's work is indicated in the final section 7.