# Factorization formulas of $K$-$k$-Schur functions I

## TL;DR

This paper introduces new factorization formulas for $K$-$k$-Schur functions, showing divisibility properties and explicit formulas for certain partitions, extending known results for $k$-Schur functions.

## Contribution

It establishes divisibility and factorization formulas for $K$-$k$-Schur functions related to multiple $k$-rectangles, generalizing previous $k$-rectangle factorizations.

## Key findings

- $g^{(k)}_{R_t}$ divides $g^{(k)}_{R_t ext{cup} ext{lambda}}$
- Explicit formulas for $g^{(k)}_{P ext{cup} ext{lambda}}/g^{(k)}_{P}$ in special cases
- Generalization of $k$-rectangle factorization to $K$-$k$-Schur functions

## Abstract

We give some new formulas about factorizations of $K$-$k$-Schur functions $g^{(k)}_{\lambda}$, analogous to the $k$-rectangle factorization formula $s^{(k)}_{R_t\cup\lambda}=s^{(k)}_{R_t}s^{(k)}_{\lambda}$ of $k$-Schur functions, where $\lambda$ is any $k$-bounded partition and $R_t$ denotes the partition $(t^{k+1-t})$ called \textit{$k$-rectangle}. Although a formula of the same form does not hold for $K$-$k$-Schur functions, we can prove that $g^{(k)}_{R_t}$ divides $g^{(k)}_{R_t\cup\lambda}$, and in fact more generally that $g^{(k)}_{P}$ divides $g^{(k)}_{P\cup\lambda}$ for any multiple $k$-rectangles $P=R_{t_1}^{a_1}\cup\dots\cup R_{t_m}^{a_m}$ and any $k$-bounded partition $\lambda$. We give the factorization formula of such $g^{(k)}_{P}$ and the explicit formulas of $g^{(k)}_{P\cup\lambda}/g^{(k)}_{P}$ in some cases, including the case where $\lambda$ is a partition with a single part as the easiest example.