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For $a:a_1\geq \cdots\geq a_c$, let $\sigma_a$ be the corresponding Schubert cycle over $Gr(c,\infty)$. We say $a$ is of depth $k$ if $a_1-a_c=k$ ($c>1$). Let $a$ and $b$ be of depth $k_1$ and $k_2$, respectively. Is the following true?

Among all Schubert cycles inside the expansion of $\sigma_a\cdot \sigma_b$, there is one whose depth is at most $\max(k_1,k_2)$.

One need to use Littlewood-Richardson rule cleverly!

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Yes. Let $(d_1, d_2, \ldots, d_c)$ be the partition obtained by sorting $(a_1+b_c, a_2+ b_{c-1}, \cdots, a_c+b_1)$ into decreasing order. The Parthasarathy-Ranga Rao-Varadarajan conjecture, now proved by Kumar (see also Knutson and Tao), has as a special case that $\sigma_d$ has nonzero coefficient in $\sigma_a \cdot \sigma_b$.

Now, the depth of $d$ is $$\max_{1 \leq i,j \leq c} (a_i+b_{c+1-i}) - (a_j + b_{c+1-j}).$$ Whenever $i<j$, we have $$(a_i+b_{c+1-i}) - (a_j + b_{c+1-j}) = (a_i - a_j) - (b_{c+1-j} - b_{c+1-i}) \leq a_i - a_j \leq k_a.$$ Similarly, whenever $i > j$, we have $$(a_i+b_{c+1-i}) - (a_j + b_{c+1-j}) \leq k_b.$$

By the way, the PRV conjecture can be used to give an alternate proof of the result you asked about here, that the smallest possible value for $d_1$ is $\max(a_1+b_c, a_2+b_{c-1}, \ldots, a_c+b_1)$.

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