# integral schur function over standard simplex

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 Schur 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 asymptotic behaviour of the integration,

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

• An impractical idea (perhaps a totally useless one): write the Schur polynomials as a sum of monomial symmetric functions, then integrate those using "standard" methods, and conclude ... – Suvrit Jul 24 '15 at 21:46
• Perhaps changing order of integration based on the multiple contour formula here can be useful: mathoverflow.net/questions/227350/… – John Jiang Jan 2 '16 at 5:19