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.
Categories: math.CO