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Transition of Grothendieck polynomials. (English) Zbl 1052.14059
Kirillov, A. N. (ed.) et al., Physics and combinatorics. Proceedings of the Nagoya 2000 2nd international workshop, Nagoya, Japan, August 21–26, 2000. Singapore: World Scientific (ISBN 981-02-4642-0/hbk). 164-179 (2001).
From the paper: Given a vector bundle $$V$$ of rank $$n$$, the Grothendieck ring of classes of vector bundles of the relative flag manifold $${\mathcal F}(V)$$ is generated by the classes $$a_1,\dots,a_n$$ of the so-called tautological line bundles on $${\mathcal F}(V)$$. The structure sheaf of a Schubert variety in $${\mathcal F}(V)$$, having a finite resolution by vector bundles, can be expressed as a (Laurent) polynomial in the $$a_i$$, $$1/a_i$$. Explicit representatives $$G_\sigma$$, $$\sigma\in{\mathfrak S}_n$$ were defined by the author and M.-P. Schützenberger [in: Invariant theory. Lect. Notes Math. 996, 118–144 (1983; Zbl 0542.14031) and C. R. Acad. Sci., Paris, 295, 629–633 (1982; Zbl 0542.14030)] under the name “Grothendieck polynomials”. We describe how general Grothendieck of polynomials are related to those for Grassmann manifolds, which themselves are deformations of Schur functions. The geometry of Grassmann varieties is well understood thanks to Schubert, Giambelli and their successors. It is hoped that relating the Schubert subvarieties of a flag manifold to those of a Grassmannian will be of some help to understand those varieties, the singularities of which we still do not know how to describe.
For the entire collection see [Zbl 0964.00051].

MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 05E05 Symmetric functions and generalizations 05E10 Combinatorial aspects of representation theory