[Paper Review] A Pieri rule for Demazure characters of the general linear group

Sami Assaf, Danjoseph Quijada|arXiv (Cornell University)|Aug 22, 2019

Advanced Combinatorial Mathematics22 references4 citations

TL;DR

This paper presents a cancellation-free, multiplicity-free combinatorial formula for the key polynomial expansion of the product of an arbitrary key polynomial with a single-part key polynomial in the general linear group, generalizing the classical Pieri rule. The proof uses a novel insertion algorithm on Kohnert diagrams that extends the Robinson–Schensted–Knuth correspondence, with signs arising from non-disjoint image unions rather than sign-reversing involutions.

ABSTRACT

The Pieri rule is a nonnegative, multiplicity-free formula for the Schur function expansion of the product of an arbitrary Schur function with a single row Schur function. Key polynomials are characters of Demazure modules for the general linear group that generalize the Schur function basis of symmetric functions to a basis of the full polynomial ring. We prove a nonsymmetric generalization of the Pieri rule by giving a cancellation-free, multiplicity-free formula for the key polynomial expansion of the product of an arbitrary key polynomial with a single part key polynomial. Our proof is combinatorial, generalizing the Robinson--Schensted--Knuth insertion algorithm on tableaux to an insertion algorithm on Kohnert diagrams.

Motivation & Objective

• To generalize the classical Pieri rule for Schur polynomials to the setting of key polynomials, which are Demazure characters for the general linear group.
• To provide a cancellation-free, multiplicity-free formula for the product of an arbitrary key polynomial with a single-part key polynomial.
• To develop a combinatorial insertion algorithm on Kohnert diagrams that generalizes the RSK correspondence.
• To characterize when the key polynomial expansion of such a product is nonnegative, using the structure of the insertion image.

Proposed method

• The authors define a new insertion algorithm on Kohnert diagrams that generalizes the RSK insertion to the nonsymmetric setting.
• They introduce a left swap order on Kohnert diagrams to control the insertion process and ensure well-definedness.
• The main bijection is constructed via three key maps: bottom insertion, rectification, and top insertion, which are proven to be inverse bijections.
• The insertion process tracks the effect of multiplying a key polynomial by a single-row Schur function, with signs arising from overlapping image components.
• The proof relies on a stratification of Kohnert diagrams based on the insertion process, with injective stratum maps and image characterization.
• The method avoids sign-reversing involutions by directly analyzing the structure of the insertion image, which may not be disjoint.

Experimental results

Research questions

• RQ1 How can the classical Pieri rule for Schur polynomials be generalized to the nonsymmetric setting of key polynomials?
• RQ2 What combinatorial insertion algorithm on Kohnert diagrams realizes the structure constants in the key polynomial expansion of a product with a single-part key polynomial?
• RQ3 Why do signs appear in the expansion, and when is the expansion actually nonnegative?
• RQ4 How does the image of the insertion map relate to the structure of the resulting key polynomial expansion?
• RQ5 Can this insertion-based approach provide new insight into Schubert structure constants or Littlewood–Richardson rules for key polynomials?

Key findings

• The paper establishes a cancellation-free, multiplicity-free formula for the product of an arbitrary key polynomial and a single-part key polynomial, with coefficients in {1, -1}.
• The insertion algorithm on Kohnert diagrams generalizes the RSK correspondence and provides a direct combinatorial proof of the nonsymmetric Pieri rule.
• The image of the insertion map is a union of Kohnert diagrams that is not necessarily disjoint, explaining the appearance of signs in the expansion.
• The authors characterize when the key expansion is nonnegative: precisely when the image of the insertion map is disjoint, which occurs under specific combinatorial conditions on the input diagrams.
• The result provides a new, simplified proof of the nonnegativity of the key expansion in the vexillary case, and offers geometric motivation via Schubert polynomials.
• The framework suggests a path toward a full Littlewood–Richardson rule for key polynomials, potentially informing the long-standing problem of computing Schubert structure constants.