A Pieri-type formula for the \({K}\)-theory of a flag manifold. (English) Zbl 1111.14047
For a flag variety \(\text{Fl}_n\), the classes of structure sheaves of Schubert varieties form an integral basis in the Grothendieck ring. A major open problem in the modern Schubert calculus is to determine the \(K\)-theory Schubert structure constants, which express the product of two Schubert classes in terms of this basis.
The authors derive explicit Pieri-type formulae in the Grothendieck ring of a flag variety, which generalize both the \(K\)-theory Monk formula [see C. Lenart, J. Pure Appl. Algebra 179, No. 1–2, 137–158 (2003; Zbl 1063.14060)] and the cohomology Pieri formula [see F. Sottile, Ann. Inst. Fourier 46, No. 1, 89–110 (1996; Zbl 0837.14041)]. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special classes are indexed by cycles of the form \((k-p+1, k-p+2,\dots,k+1)\) or \((k+p, k+p-1, \dots ,k)\), and are pulled back from the projection of \(\text{Fl}_n\) to the Grassmannian of \(k\)-planes.
The formula is expressed in terms of certain labelled chains in the \(k\)-Bruhat order of the symmetric group, and the multiplicities in it are certain binomial coefficients. The proof exploits algebraic-combinatorial setting of Grothendieck polynomials and a Monk-like formula for multiplying a Grothendieck polynomial by a variable.
The authors derive explicit Pieri-type formulae in the Grothendieck ring of a flag variety, which generalize both the \(K\)-theory Monk formula [see C. Lenart, J. Pure Appl. Algebra 179, No. 1–2, 137–158 (2003; Zbl 1063.14060)] and the cohomology Pieri formula [see F. Sottile, Ann. Inst. Fourier 46, No. 1, 89–110 (1996; Zbl 0837.14041)]. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special classes are indexed by cycles of the form \((k-p+1, k-p+2,\dots,k+1)\) or \((k+p, k+p-1, \dots ,k)\), and are pulled back from the projection of \(\text{Fl}_n\) to the Grassmannian of \(k\)-planes.
The formula is expressed in terms of certain labelled chains in the \(k\)-Bruhat order of the symmetric group, and the multiplicities in it are certain binomial coefficients. The proof exploits algebraic-combinatorial setting of Grothendieck polynomials and a Monk-like formula for multiplying a Grothendieck polynomial by a variable.
Reviewer: Dmitri Panyushev (Moskva)
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 05E99 | Algebraic combinatorics |
| 19L64 | Geometric applications of topological \(K\)-theory |
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