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

Abstract

We give some new formulas about factorizations of $K$-$k$-Schur functions $g_{\lambda }^{(k)}$, analogous to the $k$-rectangle factorization formula $s_{R_t\cup \lambda }^{(k)}=s_{R_t}^{(k)}{s}_{\lambda }^{(k)}$ 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 $g_{R_t}^{(k)}$ divides $g_{R_t\cup \lambda }^{(k)}$, and in fact more generally that $g_P^{(k)}$ divides $g_{P\cup \lambda }^{(k)}$ for any multiple $k$-rectangles $P=R_{t_1}^{a_1}\cup \cdots \cup R_{t_m}^{a_m}$ and any $k$-bounded partition $\lambda$. We give the factorization formula of such $g_P^{(k)}$ and the explicit formulas of $g_{P\cup \lambda }^{(k)}/g_P^{(k)}$ in some cases, including the case where $\lambda$ is a partition with a single part as the easiest example.

arXiv comment36 pages