Three formulas for CSM classes of open quiver loci

In the space of equioriented type $A$ quiver representations, we define subvarieties called “open quiver loci” by placing strict rank conditions on the maps within representations. The closures of these subvarieties are the quiver loci, whose equivariant cohomology classes are the quiver polynomials of Buch and Fulton. We present one geometric formula and two combinatorial formulas that compute equivariant Chern-Schwartz-MacPherson (CSM) classes of open quiver loci; these classes refine the data of the quiver polynomials. The second combinatorial formula is in terms of “chained generic pipe dreams,” which modify the pipe dreams of Bergeron and Billey to more strongly resemble the lacing diagrams of Abeasis and Del Fra. We also present two new formulas for quiver polynomials; these are streamlined versions of known formulas due to Knutson, Miller, and Shimozono, in the sense that they contain fewer terms.

💡 Research Summary

The paper studies representations of the equioriented type‑A quiver 0→1→⋯→n and introduces “open quiver loci” Ω⁰_r defined by imposing exact rank conditions r_{ij} on each composite map V_i→V_j. These open loci are single GL(V₀)×⋯×GL(V_n)‑orbits; their closures Ω_r are the classical quiver loci defined by weak (≤) rank conditions, whose equivariant cohomology classes are the well‑known quiver polynomials of Buch–Fulton. The authors aim to compute the equivariant Chern–Schwartz–MacPherson (CSM) classes of the open loci, which refine quiver polynomials by also encoding the topological Euler characteristic.

The first main result (Theorem 5.4) expresses the CSM class of Ω⁰_r as a ratio of CSM classes of Schubert cells in the complete flag variety. Using the Zelevinsky map Z: Hom(V₀,…,V_n) → GL_d (with d = Σ dim V_i), each quiver locus Ω_r is identified with the intersection X_{z(r)} ∩ X_{w₀w₀} of a Schubert variety X_{z(r)} and a fixed opposite Schubert cell X_{w₀w₀}. Since the CSM classes of Schubert cells are known (Aluffi–Mihalcea), the ratio yields an explicit polynomial in the equivariant Chern roots.

The second formula (Proposition 5.5) translates the ratio into a sum over Bergeron–Billey pipe dreams. A pipe dream is a d×d grid filled with crosses and elbows; crosses correspond exactly to the 1‑entries of the Zelevinsky permutation z(r). Assigning the weight (x_i−y_j) to each cross and summing over all pipe dreams compatible with z(r) reproduces the CSM class of Ω⁰_r. This recovers the KMS06 ratio formula in a purely combinatorial guise.

The third and most novel contribution (Theorem 5.12) introduces “chained generic pipe dreams” (CGPDs). These are pipe‑dream‑like configurations that satisfy a genericity condition (no two pipes intersect) and a chaining condition that mirrors the lacing diagrams of Abeasis–Del Fra. Each chain starts and ends at positions dictated by the rank array r, and the weight of a CGPD is again a product of (x_i−y_j) over its crosses. By applying Su’s CSM‑Schubert weight theory, the authors show that the sum of CGPD weights equals the CSM class of the open quiver locus. Specializing the equivariant parameters (x_i = y_i) collapses the formula to a new, compact expression for the ordinary quiver polynomial (Corollary 5.14).

In Section 3 the authors revisit the four KMS06 formulas for quiver polynomials and present streamlined versions of two of them. By exploiting the block structure of the Zelevinsky permutation they eliminate many redundant terms, yielding formulas with significantly fewer summands and thus improved computational efficiency.

Overall, the paper bridges equivariant CSM theory with quiver geometry, providing three complementary perspectives—geometric (Schubert cell ratio), combinatorial (pipe dreams), and refined combinatorial (chained generic pipe dreams)—for the same invariant. The work not only refines known quiver polynomials but also offers practical tools for explicit calculations, and suggests avenues for extending these techniques to other quiver types or to more general group actions.