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Lowering operators on $K$-$k$-Schur functions and a lowering operator formula for closed $K$-$k$-Schur functions

Yaozhou Fang, Xing Gao, Li Guo

TL;DR

The paper develops a systematic framework for lowering operators on $K$-$k$-Schur functions, culminating in a reductive formula for the operator $L_z$ and a concrete lowering-operator expression for closed $K$-$k$-Schur functions. By embedding $K$-$k$-Schur functions into the broader theory of Katalan functions and using Mirror Lemmas, the authors obtain a precise expansion of lowerings in terms of other $K$-$k$-Schur functions, organized by root-ideal structure and Bruhat order. A key result is the closed-form formula $ ilde{ rak g}_{oldsymbol{\lambda}}^{(k)} = (1-G_{1}^{ot})ig( extstyle\sum_{oldsymbol{ u}, w_{oldsymbol{ u}} le w_{oldsymbol{\lambda}}} g_{oldsymbol{ u}}^{(k)}ig)$, which provides an operator-theoretic route to the closed $k$-Schur Katalan function. This leads to a combinatorial proof of a conjecture on closed $k$-Schur Katalan functions, connecting to the $K$-theoretic Schubert calculus of the affine Grassmannian and offering new insights into the interplay between lowering operators, Bruhat order, and symmetric-function realizations.

Abstract

This paper gives a systematic study of the lowering operators acting on the $K$-$k$-Schur functions, motivated by the pivotal role played by the operators in the definition and study of Katalan functions. A lowering operator formula for closed $K$-$k$-Schur functions is obtained. As an application, a combinatorial proof is provided to a conjecture on closed $k$-Schur Katalan functions, posed by Blasiak, Morse and Seelinger, and recently proved by Ikeda, Iwao and Naito by a different method.

Lowering operators on $K$-$k$-Schur functions and a lowering operator formula for closed $K$-$k$-Schur functions

TL;DR

The paper develops a systematic framework for lowering operators on --Schur functions, culminating in a reductive formula for the operator and a concrete lowering-operator expression for closed --Schur functions. By embedding --Schur functions into the broader theory of Katalan functions and using Mirror Lemmas, the authors obtain a precise expansion of lowerings in terms of other --Schur functions, organized by root-ideal structure and Bruhat order. A key result is the closed-form formula , which provides an operator-theoretic route to the closed -Schur Katalan function. This leads to a combinatorial proof of a conjecture on closed -Schur Katalan functions, connecting to the -theoretic Schubert calculus of the affine Grassmannian and offering new insights into the interplay between lowering operators, Bruhat order, and symmetric-function realizations.

Abstract

This paper gives a systematic study of the lowering operators acting on the --Schur functions, motivated by the pivotal role played by the operators in the definition and study of Katalan functions. A lowering operator formula for closed --Schur functions is obtained. As an application, a combinatorial proof is provided to a conjecture on closed -Schur Katalan functions, posed by Blasiak, Morse and Seelinger, and recently proved by Ikeda, Iwao and Naito by a different method.

Paper Structure

This paper contains 15 sections, 27 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Schubert calculus and Schur functions
  3. Lowering operators and closed $K$-$k$-Schur functions
  4. Outline and notations
  5. Katalan functions and Mirror Lemmas
  6. Katalan functions
  7. Bounce graphs and Mirror Lemmas
  8. The lowering operator $L_z$ acting on the $K$-$k$-Schur function $g_\lambda^{(k)}$
  9. The lowering operator $L_{z}$ for $z\in[b_\lambda+1,\ell]$
  10. The lowering operator $L_{z}$ for $z\in[b_\lambda]$
  11. Summary on the lowering operator
  12. A combinatorial proof of a conjecture on closed $k$-Schur Katalan functions
  13. Partitions, cores and Bruhat order on $\tilde{S}_{k+1}$
  14. A lowering operator formula for closed $K$-$k$-Schur functions
  15. The proof of Theorem \ref{['th:aim-1st']}

Key Result

Theorem 1.1

$($=Theorem thm:closedK-k-Schur$)$ Let $\lambda\in{\rm P}_{\ell}^{k}$. Then Here we denote $L_{S}:=\prod_{z\in S}L_{z}$ for a multiset $S$ on $[\ell]$.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 2.7
  • ...and 52 more