Cotangent Schubert calculus is an extension of Schubert calculus, a classical topic in enumerative geometry, that builds upon recent advances in algebraic geometry as well as developments in geometric representation theory and quantum integrable systems.
The author presents the Macaulay2 package CotangentSchubert, which is concerned with cohomology or \(K\)-theory of the cotangent bundle \(T^*(P\backslash G)\) of the partial flag variety \(P\backslash G\). Nonequivariantly, \(H^*(P\backslash G)\) (similarly in \(K\)-theory), and one always include equivariance with respect to scaling of the fiber of the cotangent bundle, i.e., \(H^*_{\mathbb{C}^{\times}}(P\backslash G) \cong H^*(P\backslash G)[h]\) (in \(K\)-theory, \(K_{\mathbb{C}^{\times}}(P\backslash G) \cong K(P\backslash G)[t^{\pm}]\)), where \(h\) (or \(t\)) is the equivariant parameter. One objective for this package is to define the basic objects of cotangent Schubert calculus in the context of equivariant \(K\)-theory of partial flag varieties. The second objective is to give the expansion of the product of Segre classes in terms of certain combinatorial gadgets known as puzzles, a generalization of the puzzles of Knutson and Tao for ordinary Schubert calculus of Grassmannians. He shows its use in computing motivic Chern and Segre classes of Schubert cells of partial flag varieties and in checking recently found combinatorial formulae for their products.