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,

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