A formalism for equivariant Schubert calculus
type: Item
instance of: scholarly article
title: A formalism for equivariant Schubert calculus
zbMATH Open document ID: 1186.14057
author: Dan Laksov
published in: Algebra \& Number Theory
publication date: 2010-01-05T00:00:00Z
review text: 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 …
reviewed by: Letterio Gatto
Mathematics Subject Classification ID: 14N15
Mathematics Subject Classification ID: 57R91
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 5652989
zbMATH Keywords: equivariant Pieri's formula
zbMATH Keywords: equivariant Giambelli's formula
MaRDI profile type: Publication
full work available at URL: ant.2009.3.711
OpenAlex ID: W2059298005
DOI: 10.2140/ANT.2009.3.711
description: scientific article; zbMATH DE number 5652989