[Paper Review] Padded Schubert polynomials and weighted enumeration of Bruhat chains

Christian Gaetz, Yibo Gao|arXiv (Cornell University)|Apr 30, 2019

Advanced Combinatorial Mathematics10 references1 citations

TL;DR

This paper introduces a unified framework using padded Schubert polynomials to prove a one-parameter family of weighted enumeration identities for saturated chains in the strong Bruhat order on the symmetric group. It generalizes classical results of Stembridge (Chevalley weights) and Macdonald (reduced word identities), showing that the total weighted count of maximal chains from the identity to the longest permutation w₀ is always (n choose 2)! regardless of a parameter z, under specific weight assignments based on regions in permutation matrices. The key contribution is a common generalization of the code weight and Chevalley weight identities, with the weighted chain count independent of the parameter z.

ABSTRACT

We prove a common generalization of the fact that the weighted number of maximal chains in the strong Bruhat order on the symmetric group is ${n \choose 2}!$ for both the code weights and the Chevalley weights. We also define weights which give a one-parameter family of strong order analogues of Macdonald's reduced word identity for Schubert polynomials.

Motivation & Objective

• To unify and generalize the classical weighted enumeration results of Stembridge (Chevalley weights) and Macdonald (reduced word identity) for maximal chains in the strong Bruhat order.
• To explain the surprising coincidence between the code weight and Chevalley weight counts of maximal chains from e to w₀, both yielding (n choose 2)!.
• To construct a one-parameter family of weights on covering relations in the strong Bruhat order that generalize both the code weights and the Chevalley weights.
• To prove that the total weighted count of maximal chains from the identity permutation e to the longest permutation w₀ remains constant at (n choose 2)! for all values of the parameter z under the new weight definitions.

Proposed method

• Introduces a new weight function f(v ⋖ w) = 1 + a zA + b zB + c zC + d zD, where a, b, c, d count dots in four regions (A, B, C, D) around the transposition (i j) in the permutation matrix.
• Defines a one-parameter family of weights by specializing zA, zB, zC, zD to {0, 0, z, 2−z} as a multiset, proving the total chain count mwt(e, w₀) is independent of z.
• Uses the algebraic structure of padded Schubert polynomials and the action of divided difference operators to analyze the weighted chain counts.
• Applies the M + zR operator on Schubert polynomials, where M encodes Chevalley weights and R encodes code weight-like terms, to derive the generating function for weighted chains.
• Employs induction and principal specialization techniques to prove the invariance of the weighted chain count under parameter z.
• Leverages symmetries of the Bruhat order (e.g., inversion, left/right multiplication by w₀) to reduce the number of cases in the proof.

Experimental results

Research questions

• RQ1 Why do the weighted counts of maximal chains from e to w₀ under the code weights and the Chevalley weights both equal (n choose 2)!?
• RQ2 Can a single parameterized family of weights be constructed that generalizes both the code weights and the Chevalley weights?
• RQ3 Is the total weighted count of maximal chains from e to w₀ invariant under a one-parameter deformation of the weights?
• RQ4 Does the strong order analogue of Macdonald's reduced word identity extend to a one-parameter family of weights, with the same constant value (n choose 2)!?
• RQ5 What is the algebraic mechanism that ensures the z-invariance of the weighted chain count in the new parameterized weight families?

Key findings

• The total weighted number of maximal chains from the identity permutation e to the longest permutation w₀ is exactly (n choose 2)! for all specializations of the weight function f(v ⋖ w) = 1 + a zA + b zB + c zC + d zD where {zA, zB, zC, zD} = {0, 0, z, 2−z} as a multiset.
• The one-parameter family of weights defined by wt(v ⋖ vtij) = αi + ... + αj−1 + (bv⋖w − dv⋖w)z yields a total chain count mwt(e, w₀) = (n choose 2)! · ∏(k<ℓ)(αk + ... + αℓ−1)/(ℓ−k), which is independent of z.
• The identity mwt(w, w₀) = ( (n choose 2) − ℓ(w) )! · Sw(1, ..., 1) holds for the weight wt(v ⋖ w) = 1 + bv⋖w(2−z) + cv⋖w z, proving a strong order analogue of Macdonald's reduced word identity.
• Specializing z = 0 in the weight family recovers Stembridge's result for Chevalley weights, and specializing z = 0 in the Macdonald-type family recovers the code weight identity.
• The proof of z-invariance relies on the fact that the coefficient of Sw₀ in (M + zR)^(n choose 2) is independent of z, as shown by a combinatorial argument involving moving R operators to the right and using the vanishing of R on the constant 1.