Grothendieck classes of quiver varieties. (Q1857113)

For a sequence \(E_{\bullet}:E_{0}\to E_{1}\to \cdots \to E_{n}\) of vector bundles and bundle maps over a nonsingular variety \(X\) one can obtain a quiver variety \(\Omega_{r}=\{x\in X\mid \text{rank} (E_{i} (x) \to E_{j}(x)) \leq r_{ij}\), \(i<j\} \) where \(r=\{r_{ij}\} \) is a collection of integers with \(0\leq i<j\leq n.\) (Clearly \(\Omega_{r}\) also depends on \(E_{\bullet}\) but the notation indicating such is suppressed here.) Then \(\Omega_{r}\) is a subscheme of \(X\). The maximum codimension of \( \Omega_{r}\) is \(d(r) :=\sum_{i<j}(r_{i,j-1}-r_{ij}) (r_{i+1,j}-r_{ij})\). The objective in this paper is to find a formula for \(\mathcal{O}_{\Omega_{r}},\) the structure sheaf of \(\Omega_{r},\) in \(K^{\circ}X,\) the Grothendieck ring of algebraic vector bundles on \(X\). Denoting the class of \( \Omega_{r}\) in \(K^{\circ}X\) by \([ \mathcal{O}_{\Omega_{r}}] \) gives the following description of the formula: \[ [ \mathcal{O}_{\Omega_{r}}] =\sum_{| \mu | \geq d(r)}c_{\mu}(r) G_{\mu_{i}}(E_{1}-E_{0}) \cdots G_{\mu_{n}}(E_{n}-E_{n-1}) \] where \(c_{\mu}(r) \) are certain integers described combinatorially in the paper and \(G_{\mu_{i}}(E_{i}-E_{i-1}) \) are the stable Grothendieck polynomials (which are also defined in the paper). The sum is over a finite number of sequences of partitions \(\mu \) such that the weights sum to at least \(d(r) .\) This formula is analogous to the formula for the cohomology class of \(\Omega_{r}\) as presented in the author's previous work with Fulton, however in this situation one needs the codimension to be precisely \(d(r) .\) The paper starts with a treatment of these Grothendieck polynomials, including their geometric significance. Following, the algorithm for generating the coefficients \(c_{\mu}(r) \) is presented. The author conjectures that \((-1)^{| \mu | -d(r)}c_{\mu}(r) \geq 0,\) that is that the signs of the coefficients alternate with the weight of \(\mu .\) Evidence to support this conjecture appears in the fifth section where it is shown that Grothendieck polynomials are special cases of the formula. Finally, a Gysin formula, which calculates \(K\)-theoretic pushforwards from a Grassmann bundle, is shown using a generalization of the Jacobi-Trudi formula for Schur functions.