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 G(d,n), where $m>d$ and 1≤ d ≤ n-d-1 with a P-homogeneity degree $h(E)=d$. Particularly, we establish an upper bound of $3(n-d)-2$ for the uniform-homogeneous shreshold of G(d,n). Additionally, we construct indecomposable uniform vector bundles of rank (d+2)(n-d)+d-2+∑ᵢ₌₀ᵖd-1+p-ip-i(1+i)-p+dp that are nonhomogeneous over G(d,n).