Generic Initial Ideals And Graded Artinian Level Algebras Not Having The Weak-Lefschetz Property

We find a sufficient condition that \H is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function $\H=(h_0,h_1,..., h_{d-1}>h_d=h_{d+1})$ cannot be level if $h_d\le 2d+3$, and that there exists a level O-sequence of codimension 3 of type \H for $h_d \ge 2d+k$ for $k\ge 4$. Furthermore, we show that \H is not level if $\beta_{1,d+2}(I^{\rm lex})=\beta_{2,d+2}(I^{\rm lex})$, and also prove that any codimension 3 Artinian graded algebra $A=R/I$ cannot be level if $\beta_{1,d+2}(\Gin(I))=\beta_{2,d+2}(\Gin(I))$. In this case, the Hilbert function of $A$ does not have to satisfy the condition $h_{d-1}>h_d=h_{d+1}$. Moreover, we show that every codimension $n$ graded Artinian level algebra having the Weak-Lefschetz Property has the strictly unimodal Hilbert function having a growth condition on $(h_{d-1}-h_{d}) \le (n-1)(h_d-h_{d+1})$ for every $d > \theta$ where $$h_0<h_1<...<h_\alpha=...=h_{\theta}>...>h_{s-1}>h_s.$$ In particular, we find that if $A$ is of codimension 3, then $(h_{d-1}-h_{d}) < 2(h_d-h_{d+1})$ for every $\theta< d <s$ and $h_{s-1}\le 3 h_s$, and prove that if $A$ is a codimension 3 Artinian algebra with an $h$-vector $(1,3,h_2,...,h_s)$ such that $$h_{d-1}-h_d=2(h_d-h_{d+1})>0 \quad \text{and} \quad \soc(A)_{d-1}=0$$ for some $r_1(A)<d<s$, then $(I_{\le d+1})$ is $(d+1)$-regular and $\dim_k\soc(A)_d=h_d-h_{d+1}$.