Let $r\leq n$ and $d$ be positive integers. A *probability vector* is a vector of non-negative entries that sum to 1. For each probability vector $\lambda$ of length $n$, let


$$s(\lambda)=(\dim[\pi] \cdot s\_{\pi}(\lambda))\_{\substack{\pi \vdash d}},$$


where $\pi \vdash d$ means that $\pi$ is an integer partition of $d$, $[\pi]$ denotes the irreducible representation of $\mathfrak{S}\_d$ indexed by $\pi$, and $s\_{\pi} (\lambda)$ is the Schur polynomial indexed by $\pi$ evaluated at $\lambda$. It can be checked that $s(\lambda)$ is a probability vector.


**Question:** Consider the probability vector


$$ t = \left(\frac{1}{\sum\_{ \pi \vdash d, \ell(\pi)\leq r } s\_{\pi}(1^n)^2}\right) \big(s\_{\pi}(1^n)^2\big)\_{ \pi \vdash d, \ell(\pi)\leq r },$$


where $\ell(\pi)$ is the number of integers appearing in the integer partition $\pi$. Is it true that $t$ is in the convex hull of the $s(\lambda)$, where $\lambda$ ranges over all probability vectors containing at most $r$ non-zero entries?


The answer is trivially yes when $r=1$.


**Notes:** The dimension of $[\pi]$ can be computed using the [Hook-Length formula](https://en.wikipedia.org/wiki/Hook_length_formula) and the Schur polynomials can be found [here](https://en.wikipedia.org/wiki/Schur_polynomial).



The convex hull of Schur polynomial evaluations
If I understand the question correctly then this is false. Since you want $t$ to live in the same vector space as the $s(\lambda)$'s, I am assuming that the vector $t$ has zero coordinates at each $\pi \vdash d, \ell(\pi)> r$.


Taking $d=3, r=2, n=6$ and indexing the vectors in $\mathbb C^3$ as $(x\_{(3)}, x\_{(2,1)}, x\_{(1,1,1)})$ gives us the vectors
$$s(\lambda)=(\lambda\_1^2+\lambda\_2^2, 2\lambda\_1\lambda\_2, 0)$$
and
$$t=(\frac{16}{41}, \frac{25}{41}, 0)$$
Now, the convex hull of the $s(\lambda)$'s is given by the family of vectors $(a, 1-a, 0)$ with $\frac{1}{2}\le a\le 1$, so $t$ is not contained in this convex hull.




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All that being said, the answer is *positive* for the special case $r=n$, where the probability measure induced by $t$ is (a specialization of) a Schur measure on partitions as studied by Okounkov. This is known to induce a harmonic function on the Young graph and therefore is in the convex hull of the $s(\lambda)$'s (known as *central measures*) by the Edrei-Thoma machinery that was mentioned by Sam above in the comments.


