Let $Gr(c,\infty)$ be the complex grassmannian of $c$-dimensional subspaces of the infinite dimensional complex space. Every finite dimensional grassmannian, $Gr(c,N)$, can be thought as a subspace of $Gr(c,\infty)$.

Schubert cycles $\sigma_a$, $a: a_1\geq \cdots \geq a_c\geq 0$, are a basis for the cohomology of $Gr(c,\infty)$, and by restriction, they form a basis of cohomology of $Gr(c,N)$. The restriction $\sigma_a|_{Gr(c,N)}$ is non-zero if and only if $a_1\leq N-c$.

For every two such cycles, $\sigma_a$ and $\sigma_b$, their intersection (or product, depending on the point of view) is of the form $$\sigma_a \cdot \sigma_b =\sum n(c) \sigma_{c},$$ where $c$ runs over Shubert cycles whose degree is the sum of degrees of the left side, and $n(c)$ is some non-negative integer.

Among all $\sigma_c$ on the right such that $n(c)>0$, I am looking for the one whose $c_1$ is minimum, i.e. I am looking for the minimum $N=N(a,b)$ such that $(\sigma_a\cdot \sigma_b)|_{Gr(c,N)}$ is non-zero.

**Question**: Is there an explicit equation or a non-trivial upper bound on $N(a,b)$, in terms of $a,b$?

If yes, then

what if I have more than two terms in the product, i.e. is there an equation or upper bound for $N(a,b,c,\cdots,d)$?