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 Modular Jordan type for $k[x, y]/(x^m,y^n)$ for $m = 3, 4$ J. Korean Math. Soc.Published online August 30, 2019 Jung-Pil Park and Yong-Su Shin Seoul National University, Sungshin Women University Abstract : A sufficient condition for an Artinian complete intersection quotient $S=k[x,y]/(x^m,y^n)$, where $k$ is an algebraically closed field of a prime characteristic, to have the SLP was proved by Glasby, Praezer, and Xia in \cite{GPX}. In contrast, we find a necessary and sufficient condition on $m$ and $n$ that $S$ fails to have the SLP. Moreover, we find the Jordan type for $S$ failing to have the SLP for $m\le n$ and $m=3,4$. Keywords : Jordan types, Strong Lefschetz property, Weak Lefschetz property, Hilbert function MSC numbers : 13A02, 16W50 Full-Text :