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I'm reading the paper "Chern Classes of Schubert Cells and Varieties" by Paolo Aluffi and Leonardo Constantin Mihalcea.

I'm going through this paper rather slowly, but I'm a bit stuck on the second page. So far, I understand that the schubert varieties in the grassmanian $G_d(V)$ paramaterize the d-dimensional subspaces of V that satisfy incidence conditions with a flag of subspaces. Are these incidence conditions set or able to be computed in some way? I ask because on the second page it gives the example of $\mathbb{S}(7\geq 5 \geq 3 \geq 2)$, the schubert variety of the partition $7\geq 5 \geq 3 \geq 2$: The young diagram of the partition

The paper then goes on to say if we embed this Schubert variety in the Grassmanian $G_5(V)$ of a 13-dimensional vector space V we can realize it as the subvariety parameterizing supspaces interscting a fixed flag of subspaces of dimensions 1,4,6,9,12 in dimensions $\geq$1,2,3,4,5 respectively. I'm not sure how this follows, or where the dimensions of the flag come from. The paper says it is more common to associate this latter realization with the complementary young diagram: the complementary young diagram

I noticed a couple of things with this one. The rows and columns add up to 13 (the dimension of V), which I was wondering is this a coincidence? Another thing I noitced is that the amount of boxes (vertically) in white is the same as 1,2,3,4,5.

To sumamrize, my questions are:

  1. Are the incidence conditions with a flag of subspaces fixed or computable in some way?
  2. In the example given, where in the world do the numbers 1,4,6,9,12 come from?
  3. Is it a coincidence that the number of rows and columns add up to 13, the dimension of V?
  4. How should I intepret the young diagrams? They are new to me, and I'm not sure what information I should get from them in this context.
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