An arithmetic Schubert calculus
Let \({\mathbb{G}}(n,p)\) be the Grassmann variety of \(p\)-dimensional subspaces of an \(n\)-dimensional space over \(\text{Spec } \mathbb{Z}\). It is shown that the Chow-Arakelov ring of \({\mathbb{G}}(n,p)\) is isomorphic to the ring \({\mathcal A}(p,n)\), where \({\mathcal A}(p,n)\) is defined as follows. Let \({\mathbb{A}}=\mathbb{Z} [\widehat{a}_2,\dots, \widehat{a}_{n-p}, \widehat{b}_1,\dots, \widehat{b}_p]\) and \({\mathbb{B}}=\mathbb{R} [a_1,\dots, a_{n-p},b_1,\dots, b_p,c_2,\dots,c_n]\). Consider the ring \({\mathbb{A}}\oplus{\mathbb{B}}\) with multiplication \((\widehat{\alpha},\alpha) (\widehat{\beta},\beta)= (\widehat{\alpha} \widehat{\beta}, \omega(\widehat{\alpha})\beta+ \alpha\omega(\widehat{\beta}))\), where \(\omega:{\mathbb{A}}\to{\mathbb{B}}\) is defined by \(\omega(\widehat{a}_i)= a_i\) and \(\omega(\widehat{b}_i)= b_i\). The ring \({\mathcal A}(p,n)\) is the quotient of \({\mathbb{A}}\oplus{\mathbb{B}}\) by certain relations.
scientific article; zbMATH DE number 850805
scientific article