A positive combinatorial formula for the double Edelman--Greene coefficients

Lam, Lee, and Shimozono introduced the double Stanley symmetric functions in their study of the equivariant geometry of the affine Grassmannian. They proved that the associated double Edelman–Greene coefficients, the double Schur expansion coefficients of these functions, are positive, a result later refined by Anderson. They further asked for a combinatorial proof of this positivity. In this paper, we provide the first such proof, together with a combinatorial formula that manifests the finer positivity established by Anderson. Our formula is built from two combinatorial models: bumpless pipedreams and increasing chains in the Bruhat order. The proof relies on three key ingredients: a correspondence between these two models, a natural subdivision of bumpless pipedreams, and a symmetry property of increasing chains.

💡 Research Summary

The paper addresses a long‑standing open problem in algebraic combinatorics: to give a purely combinatorial proof of the positivity of the double Edelman–Greene coefficients, which are the double‑Schur expansion coefficients of the double Stanley symmetric functions introduced by Lam, Lee, and Shimozono. The geometric proof of Lam–Lee–Shimozono (LLS) showed that each coefficient j₍w,λ₎(y) lies in the monoid generated by differences (y_i−y_j) with i≺j, and Anderson later refined this result by imposing stricter degree constraints on three types of differences (type 1: 0<i<j, type 2: i<j≤0, type 3: j≤0<i). However, no combinatorial formula exhibiting these refined constraints was known.

The authors resolve this by constructing an explicit combinatorial formula that simultaneously displays LLS‑positivity and Anderson’s refined positivity. Their approach hinges on two combinatorial models:

1. Bumpless pipedreams (BPDs) – tilings of regions in the integer lattice ℤ×ℤ with six tile types, each contributing a factor (x_i−y_j). BPDs were previously shown to encode double Schubert polynomials and their back‑stable limits.

2. Increasing chains in Bruhat order – sequences of permutations linked by covering relations that increase a prescribed statistic; originally introduced by Sottile to model Pieri rules.

The proof proceeds in three main stages:

Stage 1 – Horizon cut.
A BPD for a given permutation w is split along the horizontal line separating rows ≤0 (upper half‑plane H↑) from rows >0 (lower half‑plane H↓). The lower half‑plane piece is further organized by a “non‑increasing” permutation σ∈S⁻. The authors define a family LBPD(σ,w) of BPDs in H↓ and introduce generating functions S⁻{σ,w}(x;y). Using the involution ω₁ (which swaps x_i with x{1−i} and similarly for y), they show that the original double Schur coefficients a_{wλ}(x;y) can be expressed as a sum over σ of ω₁‑twisted S⁻‑functions. This reduces the problem to finding a formula for S⁻_{σ,w}(y;y) that uses only type 1 and type 3 differences.

Stage 2 – Diagonal cut.
Inside H↓ the authors cut along the diagonal, separating the strictly upper triangular region A={0<i<j} from the lower triangular (trapezoid) region A’={0<i, j≤i}. The cut is governed by a boundary condition γ, which records how pipes enter and exit the region. For each γ they define a set BPD(γ) of pipedreams confined to A. The generating function S_γ(x;y)=∑_{D∈BPD(γ)}wt(D) already yields a sum of distinct products of type 1 factors after setting x→y, because every tile in A contributes a factor (y_i−y_j) with i0. Thus the remaining task is to handle the contribution from A’, which must be expressed using only type 3 factors.

Stage 3 – Bijection with increasing chains.
The authors translate the problem on A’ into the language of increasing chains. For a given γ they construct permutations U,W and an integer sequence α such that there is a weight‑preserving bijection
 BPD(γ) ↔ C(U,W,rev(α)),
where C(U,W,α) denotes the set of increasing α‑chains from U to W in Bruhat order. The weight of a chain is a product of factors (x_i−y_j) determined by the chain’s steps; after multiplying by the universal Vandermonde product ∏_{i<j}(x_i−y_j) and applying the symmetry property (Proposition 2.29) that C(u,w,α) is invariant under reversing α and the x‑variables, the authors obtain the identity

 S_γ(y;y)=∑{(u₁,…,u_n)∈C(U,W,α)} wt_n^α(u₁,…,u_n)·∏{i<j}(y_i−y_j).

Crucially, each term on the right is either zero or a distinct product of type 3 differences, and no two chains produce the same monomial. This establishes a combinatorial formula for S_γ(y;y) satisfying Anderson’s constraints.

Putting the pieces together, the authors express the original double Edelman–Greene coefficient as

 j_{wλ}(y)=∑{σ∈S⁻} ω₁( S⁻{σ,wλ′}(y;y) ) · S⁻_{σ,w}(y;y),

where each summand is a product of distinct type 1 or type 3 factors, with type 1 appearing at most once per monomial and type 3 appearing at most twice, exactly as required by Theorem 1.1 (Anderson). Hence the positivity of j_{wλ}(y) is proved combinatorially, and the formula makes the refined degree bounds manifest.

The paper concludes with acknowledgments and a brief discussion of future directions, such as algorithmic implementation of the formula, extensions to other families of symmetric functions, and deeper connections between bumpless pipedreams and Bruhat‑order combinatorics.

Overall, the work provides the first explicit combinatorial proof of the double Edelman–Greene positivity, introduces a novel correspondence between bumpless pipedreams and increasing chains, and delivers a concrete formula that exhibits the refined Anderson positivity. This bridges a gap between geometric representation theory and purely combinatorial methods in the study of double Stanley functions.