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A Murnaghan-Nakayama rule for Grothendieck polynomials of Grassmannian type

Khanh Nguyen Duc, Dang Tuan Hiep, Tran Ha Son, Do Le Hai Thuy

Abstract

We consider the Grothendieck polynomials appearing in the K-theory of Grassmannians, which are analogs of Schur polynomials. This paper aims to establish a version of the Murnaghan-Nakayama rule for Grothendieck polynomials of the Grassmannian type. This rule allows us to express the product of a Grothendieck polynomial with a power sum symmetric polynomial into a linear combination of other Grothendieck polynomials.

A Murnaghan-Nakayama rule for Grothendieck polynomials of Grassmannian type

Abstract

We consider the Grothendieck polynomials appearing in the K-theory of Grassmannians, which are analogs of Schur polynomials. This paper aims to establish a version of the Murnaghan-Nakayama rule for Grothendieck polynomials of the Grassmannian type. This rule allows us to express the product of a Grothendieck polynomial with a power sum symmetric polynomial into a linear combination of other Grothendieck polynomials.

Paper Structure

This paper contains 6 sections, 5 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric Polynomials
  4. Partitions, Diagrams and Binary Tableaux
  5. Grothendieck Polynomials of Grassmannian Type
  6. Proof of the Main Theorem

Key Result

Theorem 1.1

For any partition $\lambda$ of length at most $n$ and $k \in \mathbb{Z}_{>0}$, we have where the sum runs over all partitions $\mu$ of length at most $n$, $\mu \geq \lambda$ such that $c(\mu/\lambda) \leq k$, $\mu/\lambda$ is connected and the maximal ribbon along its northwest border has size at least $k$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Theorem 3.1
  • Lemma 3.2: Theorem 3.2, lenart2000combinatorial
  • proof : Proof of Theorem \ref{['main1']}
  • ...and 3 more