[Paper Review] Grothendieck polynomials via permutation patterns and chains in the Bruhat order

Cristian Lenart, Shawn Robinson+1 more|arXiv (Cornell University)|May 28, 2004

Advanced Combinatorial Mathematics16 references4 citations

TL;DR

This paper presents new combinatorial formulas for Grothendieck polynomials using chains in the Bruhat order and permutation patterns, establishing that coefficients in these polynomials are Schubert structure constants. It derives a Pieri-type formula in K-theory and generalizes specialization formulas, showing that every monomial coefficient in a Grothendieck polynomial arises naturally as a Schubert structure constant via geometric pullbacks in the Grothendieck ring.

ABSTRACT

We give new formulas for Grothendieck polynomials of two types. One type expresses any specialization of a Grothendieck polynomial in at least two sets of variables as a linear combination of products Grothendieck polynomials in each set of variables, with coefficients Schubert structure constants for Grothendieck polynomials. The other type is in terms of chains in the Bruhat order. We compare this second type to other constructions of Grothendieck polynomials within the more general context of double Grothendieck polynomials and the closely related H-polynomials. Our methods are based upon the geometry of permutation patterns.

Motivation & Objective

• To develop a chain-theoretic construction of Grothendieck polynomials using the Bruhat order and permutation patterns.
• To generalize specialization formulas for Grothendieck polynomials in multiple variable sets.
• To identify coefficients of monomials in Grothendieck polynomials as Schubert structure constants via geometric pullbacks in K-theory.
• To compare the new construction with existing ones, including double Grothendieck and H-polynomials.
• To establish a Pieri-type formula in K-theory that underlies the main results.

Proposed method

• The authors use geometric pullbacks in K-theory along embeddings of flag varieties to derive formulas for Grothendieck polynomials.
• They construct Grothendieck polynomials as linear combinations of products of polynomials via chains in the Bruhat order.
• A key technique involves the pattern map on permutations, which induces a pullback in the Grothendieck ring.
• The method relies on a Monk-type formula in K-theory and a dual Pieri-type formula for multiplication by special Grothendieck polynomials.
• The authors generalize specialization formulas by substituting variables from multiple sets, expressing the result as a Z-linear combination of products of Grothendieck polynomials.
• They apply insertion algorithms from Bergeron and Billey to compute structure constants in special cases where permutations have restricted descent patterns.

Experimental results

Research questions

• RQ1 How can Grothendieck polynomials be constructed combinatorially using chains in the Bruhat order?
• RQ2 What is the geometric origin of the coefficients in Grothendieck polynomials when variables are specialized?
• RQ3 How do Schubert structure constants arise naturally as coefficients in Grothendieck polynomials?
• RQ4 Can a general formula for the specialization of Grothendieck polynomials in multiple variable sets be derived?
• RQ5 In what cases does the Bergeron-Billey insertion algorithm compute the Schubert structure constants in the new formulas?

Key findings

• Every coefficient of a monomial in a Grothendieck polynomial is a Schubert structure constant, arising from the pullback of structure sheaves along the pattern map.
• The paper provides a new chain-theoretic construction of Grothendieck polynomials using the Bruhat order and permutation patterns.
• A general formula expresses any specialization of a Grothendieck polynomial in multiple variable sets as a Z-linear combination of products of Grothendieck polynomials in each set.
• The coefficients in this linear combination are identified as Schubert structure constants via pullbacks in the Grothendieck ring.
• The insertion algorithm of Bergeron and Billey computes the structure constants in Theorem 9.7 when the permutations satisfy specific descent conditions.
• The coefficient of a monomial in a Schubert polynomial equals the intersection number with a Schubert variety corresponding to the monomial’s shape, as shown in Corollary 9.10.