On uniform and nonhomogeneous vector bundles over Grassmannians

We demonstrate the existence of a uniform and nonhomogeneous vector bundle $E$ of rank $(n-d)(m+1)-1$ over Grassmannian $\mathbb{G}(d,n)$, where $m>d$ and $1\le d \le n-d-1$ with a $\mathbb{P}$-homogeneity degree $h(E)=d$. Particularly, we establish an upper bound of $3(n-d)-2$ for the uniform-homogeneous shreshold of $\mathbb{G}(d,n)$. Additionally, we construct indecomposable uniform vector bundles of rank $(d+2)(n-d)+d-2+\sum\limits_{i=0}^p\tbinom{d-1+p-i}{p-i}(1+i)-\tbinom{p+d}{p}$ that are nonhomogeneous over $\mathbb{G}(d,n)$.