A formalism for equivariant Schubert calculus
The paper under review depicts a beautiful and powerful framework to deal with the multiplicative structure of the equivariant cohomology ring of Grassmannians. It contains also some foundational material intended for the full appreciation of another important paper on the same subject by the same author, ``Schubert Calculus and the Equivariant Cohomology of Grassmannians'' [Adv. Math. 217, No. 4, 1869--1888 (2008; Zbl 1136.14042)]. The latter, due to the laws of the editorial world, has already appeared two years ago, reversing both the logical and the chronological order. There are many reasons, however, to be happy for the publication, at last, of both papers: To speak frankly and with a direct language, the reviewer's opinion is that these are two important pieces of mathematics, showing that the depth of the results is not necessarily related with the technical sophistication used to get them. One should also add that these two papers are a sort of crowning of a relevant mathematical path the author began to walk a few years ago together with \textit{A.~Thorup} [Indiana Univ. Math. J. 56, No. 2, 825--845 (2007; Zbl 1121.14045) and Indiana Univ. Math. J. 58, No. 1, 283--300 (2009; Zbl 1198.14052)]. The approach taken in the paper under review makes truly clear that \textsl{equivariant Schubert calculus}, rather than being an extension of the classical theory is, instead, just a particular case of the classical picture. This becomes clear once one rephrases it via a framework, elaborated by Laksov and Thorup, based on the symmetric structure of the exterior powers of (quotients of) polynomial rings. Emphasizing the novelties of this approach, also through the comparison with pre-existent literature on the subject, will be the goal of the remaining part of this review. Recall that classical Schubert calculus is concerned with the product structure of the intersection ring of Grassmannians, which are finite \({\mathbb Z}\)-modules freely generated by the so-called \textsl{Schubert cycles}: the task consists in determining the structural constants with respect to such a basis. The goal is achieved by means of the celebrated Pieri's and Giambelli's formulas: as the intersection ring is generated as a \({\mathbb Z}\)-algebra by certain \textsl{special Schubert cycles}, the former computes the structural constants of the product of any Schubert cycle with a special one and the latter expresses any Schubert cycle as an explicit polynomial in the algebra generators. The picture can be easily generalized to the intersection theory of Grassmann bundles. Laksov and Thorup show that the Chow group of a Grassmann scheme, thought of as a module over the Chow ring (Poincaré duality), is related with an amazing algebraic fact. Let \(A[X]\) be the polynomial ring in one indeterminate, \({\mathtt p}\in A[X]\) a monic polynomial (or zero), and \(R=A[X]/{\mathtt p}\). Then the \(k\)th exterior power of \(R\) possesses a unique module structure over \(S:=A[X_1,\dots, X_n]^{sym}\), the ring of symmetric polynomials, such that the natural projection \(\otimes^kR\rightarrow \bigwedge^kR\) is \(S\)-linear. This is the \textsl{symmetric structure} of \(\bigwedge^kR\). Moreover, the latter is generated by the class modulo \({\mathtt p}\) of \(X^{k-1}\wedge X^{k-2} \wedge \dots\wedge 1\) as an \(S\)-module (and is a free \(S\)-module if \({\mathtt p}=0\)). Pieri's formula comes out from cancellations, due to the \({\mathbb Z}_2\)-symmetry of the exterior algebra, of the module multiplication of an element of \(S\) with an element of \(\bigwedge^kR\). Furthermore if \(f_1,\dots, f_k\in A[X]\), a very general Giambelli's type formula expresses the product \(f_1(X)\wedge \dots\wedge f_k(X)\) as an \(S\)-multiple of \(X^{k-1}\wedge X^{k-2}\wedge \dots\wedge X\wedge 1\). The main point is that this framework can be applied verbatim also in the case of the (small quantum) equivariant cohomology of Grassmannians. To be more precise, if \({\mathbb P}^{n-1}\) is acted on diagonally by an \(n\)-dimensional torus, there is an obvious induced action on the Grassmannian \(G(k,n)\). Many important papers came to be concerned with \textsl{equivariant versions of Pieri's or Giambelli's formulas} -- see e.g [\textit{A. Knutson} and \textit{T. Tao}, Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)] and [\textit{V. Lakshmibai, K. N. Raghavan} and \textit{P. Sankaran}, Pure Appl. Math. Q. 2, No. 3, 699--717 (2006; Zbl 1105.14065)]. Such formulas are supposed to be \textsl{equivariant corrections} of classical Pieri's and Giambelli's ones. Indeed, equivariant classes of Schubert varieties are identified with a list of polynomials satisfying certain (GKM) conditions, topologically established by \textit{M. Goresky, R. Kottwitz} and \textit{R. MacPherson} [Invent. Math. 131, No.1, 25--83 (1998; Zbl 0897.22009)]. In the paper under review the author shows that such conditions naturally arise from purely algebraic manipulations. In fact, Theorem 3.4, the central result tying the \textsl{general Schubert calculus} with the above way of spelling equivariant Schubert calculus, shows that there is a natural \(A\)-algebra isomorphism between the ring \(A[T_1,\dots, T_l]^{sym}\) of symmetric polynomials in \(l\) indeterminates and the \(A\)-algebra \(H(l)\) of all \(l\)-tuples of homogeneous polynomials, with respect to the componentwise sum and product, satisfying the GKM conditions. In other words the underlying topology of the GKM conditions has a strong algebraic root. If \(\xi\) is the equivariant first Chern class of the bundle \(O_{{\mathbb P}^{n-1}}(1)\), \(A:={\mathbb Z}[y_1,\dots,y_n]\) is the equivariant cohomology of a point and \({\mathtt p}\in A[X]\) is the monic polynomial of degree \(n\) such that \({\mathtt p}(\xi)=0\), then the general Schubert Calculus by Laksov and Thorup gives exactly, in this situation, the equivariant Schubert calculus for the Grassmannian \(G(k,n)\) Laksov's paper culminates in fact (Theorem 4.5) with the computation of what in previous literature is referred to as an ``equivariant correction'' of Pieri's formula: indeed, it is the ``classical'' Pieri written in the basis of \(A[X]\) given by the factorial Schur functions as in [\textit{L. C. Mihalcea}, Adv. Math. 203, No. 1, 1--33 (2006; Zbl 1100.14045)]. Furthermore, if one chooses the natural basis \(X^i+{\mathtt p}\) of \(A[X]/{\mathtt p}\), then Giambelli's formula has no equivariant correction while Theorem 4.5 (equivariant Pieri's formula) answers to a question posed by Lakshmibai, Raghavan and Sankaran \textsl{(loc.cit.)}. Laksov's theory, and the naturality with the GKM conditions come into play, shows not only what one should mean by Pieri's and Giambelli's formulas in Grassmann's cohomologies, but also that it is very suited for computations. Indeed, the reviewer quite disagrees with Laksov, as well as with others authors making similar claims, when he says, in the introduction, that the explicit Pieri's formula proved in Theorem 4.7 can be inferred as a particular case from the Pieri type formulas got in [\textit{S. Robinson}, J. Algebra 249, No. 1, 38--58 (2002; Zbl 1061.14060)]. The latter, clearly, are very important and have a strong theoretical impact; but it is hard even to guess, in the case of Grassmannians, how a full set of equivariant Pieri's formulas could look like or can be deduced by them. In the absence of any new reference, or without doing explicitly the job, any claim in this sense is just the vague consideration that, in mathematics, everything is related with everything else. Which, of course, is true. In conclusion, it would really be a big surprise if, in the nearest future, the reviewed paper would not put itself among the unavoidable references for people working in equivariant cohomology of homogeneous varieties, especially Grassmannians, attracting all the attention it truly deserves.
scientific article; zbMATH DE number 5652989