×

Grothendieck polynomials and quiver formulas. (English) Zbl 1084.14048

Let \(X\) be a smooth complex variety and let \(E_1\to \cdots \to E_{n-1} \to E_n\to F_n \to F_{n-1}\to \cdots \to F_1\) be a sequence of vector bundles and morphism over \(X\) such that \(\text{rank}(F_i) =\text{rank} (E_i)\) for \(1\leq i\leq n\). For any permutation \(w\in S_{n+1}\) let \(\Omega_w\) be the degeneracy locus \(\{x\in X| \text{rank}(E_q(x)\to F_p(x)) \leq r_w(p,q), \forall 1 \leq p,q\leq n\}\), where \(r_w(p,q)\) is the number of \(i\leq p\) such that \(w(i) \leq q\). W. Fulton [Duke Math. J. 96, No. 3, 575–594 (1999; Zbl 0981.14022)] gave a formula for the cohomology class of \(\Omega_w\) in \(H^\ast(X,\mathbb Z)\) as a universal Schubert polynomial in the Chern classes of the vector bundles involved, when the maps are sufficiently general.
The main result of the present article is:
For \(w\in S_{n+1}\) we have \[ [\mathcal O_{\Omega_w}] =\sum (-1)^{\l(u_1\cdots u_{2n-1}w)} G_{u_1}(E_2-E_1)\cdots G_{u_n}(F_n-E_n)\cdots G_{u_{2n-1}}(F_1-F_2) \] in \(K(X)\), where the sum is over all factorizations \(w=u_1\cdots u_{2n-1}\) in the degenerate Hecke algebra such that \(u_i \in S_{\min(i,2n-1)+1}\) for each \(i\), and \(G_u(E-E')\) is the stable Grothendieck polynomial of the permutation \(u\).
This result generalizes a previous result of the authors [Duke Math. J. 122, No. 1, 125–143 (2004; Zbl 1072.14067)].
A. Buch [Duke Math. J. 115, No. 1, 75–103 (2002; Zbl 1052.14056)] proved the quiver formula \[ [\mathcal O_{\Omega_w}] =\sum_\lambda c_{w,\lambda}^{(n)} G_{\lambda^1}(E_2-E_1)\cdots G_{\lambda^n}(F_n-E_n)\cdots G_{\lambda^{2n-1}} (F_1-F_2) \] where the sum is over finitely many sequences of partitions \(\lambda=(\lambda^1,\dots, \lambda^{2n-1})\) and where the \(c_{w,\lambda}^{(n)}\) are quiver coefficients and \(G_\alpha =G_{w_\alpha}\) is the stable Grothendieck polynomial for the Grassmannian permutation \(w_\alpha\) corresponding to \(\alpha\). The main result of the present article, together with a result of A. Lascoux [Transition on Grothendieck polynomials. Physics and Combinatorics, 2000 (Nagoya), World Sci. Publishing, River Edge, HJ, 2001, 164–179)], proves that these coefficients have alternating signs. In fact, define integers \(a_{w,\beta}\) such that \(G_w=\sum a_{w,\beta}G_\beta\) where the sum is over all permutations \(\beta\). Then the main result is equivalent to the explicit combinatorial formula for quiver coefficients: \[ c_{w,\lambda}^{(n)} =(-1)^{| \lambda| -\l(w)} \sum_{u_1\cdots u_{2n-1}=w} | a_{u_1,\lambda^1} a_{u_2,\lambda^2}\cdots a_{u_{2n-1},\lambda^{2n-1}}|. \] The proof of the main result is based on a special case of this formula proved by A. Buch [loc. cit.] together with a Cauchy identity given by A. N. Kirillov [J. Math. Sci., New York 121, No. 3, 2360–2370 (2004); translation from Zap. Nauchn. Semin. POMI 283, 123–139 (2001; Zbl 1063.05134)].
As a consequence of their result the authors obtain new formulas for the double Grothendieck polynomials of A. Lascoux and M.-P. Schützenberger [C. R. Acad. Sci., Paris, Sér. I 295, 629–633 (1982; Zbl 0542.14030)].

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14N15 Classical problems, Schubert calculus
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
05E10 Combinatorial aspects of representation theory
19E08 \(K\)-theory of schemes
PDFBibTeX XMLCite
Full Text: DOI arXiv