Combinatorial description of general double Grothendieck structure constants

Establish a combinatorial description of the structure constants C_{u,v}^w({y}) in products of double Grothendieck polynomials for arbitrary permutations u and v, including a characterization of when these structure constants are nonzero.

Background

Double Grothendieck polynomials form a basis whose multiplication is governed by structure constants C_{u,v}w({y}). Explicit combinatorial rules are known in special cases, including Grassmannian permutations with a common descent and certain two-step flag varieties, but the general multiplication problem is substantially less understood.

The unresolved problem is to provide a combinatorial description for arbitrary u and v. Such a description would also help determine which structure constants are nonzero; the paper addresses related support and stability questions but does not produce a full combinatorial rule for the coefficients themselves.

References

However, for general $u$ and $v$, it remains a longstanding open problem to give a combinatorial description of $C_{u,v}w({y})$, and relatively little is known even about which structure constants are nonzero.

— Stability of products of double Grothendieck polynomials  (2501.14691 - Hardt et al., 24 Jan 2025) in Introduction, paragraph beginning “However, for general u and v”