Let $T^d$ be the standard simplex, $$ T^d = \left\{(t_1,\cdots,t_d)\in\mathbb{R}^{d}\mid\sum_{i = 1}^{d}{t_i} = 1 \mbox{ and } t_i \ge 0 \mbox{ for all } i\right\} $$

For any partition $\lambda\vdash n$,The Shur function is defined

$$ \displaystyle s_\lambda(x_1, \ldots, x_d) = \frac{\det\Bigl(x_i^{d + \lambda_j -j}\Bigr)_{ij}}{\det\Bigl(x_i^{d-j}\Bigr)_{ij}}. $$

I would like to ask the value of the following integration, and the asymptotocal behaviour of the integration,

$$ \int _{T^d} ~~s_\lambda(t) dt $$