Let $\Delta$ be the diagonal in $\mathbb{P}^r\times\cdots\times\mathbb{P}^r$ ($k$ times), that is, $\Delta=\mbox{Proj}S/I$, where $S$ is the multi-graded k-algebra $S=k[X_0^1,\cdots,X_r^1,\cdots, X_0^k,\cdots,X_r^k]$ and $I=\langle X_i^{a}X_j^b-X_j^aX_i^b\mid \mbox{ for } 0\leq a,b\leq k \mbox{ and } 0\leq i,j\leq r \rangle$.
I know that the multigraded Hilbert Polynomial of $\Delta$ is $H_{\Delta}(d_1,\cdots,d_k)={d_1+\cdots d_k+r\choose r}$.
I've never saw a proof of this fact and I've tried to use combinatorial commutative algebra as an attempt to show this, but I couldn't.
Maybe there is some exact sequence which solve this.
I would be happy to be helped with any idea or solution. Thank you!
It is enough to note that $$ \mathcal{O}(d_1,\dots,d_k)\vert_\Delta \cong \mathcal{O}(d_1 + \dots + d_k). $$ Therefore $H_\Delta(d_1,\dots,d_k) = H_{\mathbb{P}^r}(d_1 + \dots + d_k)$.