Enumerative Real Algebraic Geometry: The manifold of partial flags

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5.iii.b. The manifold of partial flags

We use the definitions of Section 4.iii.c. Let b:=0<b₁<...<b_r<b_r+1=n be a sequence of integers. The straightforward generalization of Conjecture 5.1 for the The manifold of partial flags in n-space Fl_b is false^§.

For a in C_{n,b_i}, we have the Grassmannian Schubert variety

Z_a,i F. := {E. in Fl_b | E_{a_i} lies in the Schubert subvariety Y_aF. of Gr(a_i,n)} .

For example, if |a|=1, then Z_a,i F. is the simple Schubert variety Y_{a_i} F. of Section 4.iii.c.

Example 5.11 Let b = 2<3<5 so that Fl_b is the manifold of partial flags E₂ in E₃ in 5-space. This flag manifold has two simple Schubert varieties

Y₂ F. = { E. | E₂ meets F₃ non-trivially }

Y₃ F. = { E. | E₃ meets F₂ non-trivially }

A calculation [S08, Example 2.5] shows that the intersection of Schubert varieties

Y₂ F.(-8), Y₃ F.(-4), Y₂ F.(-2), Y₃ F.(-1), Y₂ F.(1), Y₃ F.(2), Y₂ F.(4), Y₃ F.(8)

(5.10)

is transverse with none of its 12 points real.

Thus the straightforward generalization of Conjecture 5.1 is completely false. On the other hand, if t₁ < t₂ < ... < t₈ are any of the 24,310 subsets of eight numbers from

{-6, -5, -4, -3, -2, -1, 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29} ,

then the intersection of the Schubert varieties

Y₂ F.(t₁), Y₂ F.(t₂), Y₂ F.(t₃), Y₂ F.(t₄), Y₃ F.(t₅), Y₃ F.(t₆), Y₃ F.(t₇), Y₃ F.(t₈)

(5.11)

is transverse with all of its 12 points real [So8, Example 2.5].

We have used more than 2.1 times 10⁷ seconds of CPU time investigating this problem of intersections of Schubert varieties in manifolds of partial flags given by flags osculating the rational normal curve, and a picture is emerging of what to expect, at least for Grassmannian Schubert varieties.

Suppose we have a list of indices of Grassmannian Schubert varieties

(a¹, i₁), (a², i₂), ..., (a^s, i_s) with a^j in C_{n,i_j}

where |a¹|+ |a²|+ ... + |a^s| is equal to the dimension of Fl_b. Call such a list Grassmannian Schubert data for Fl_b. We associate a composition c=(c₁, c₂, ..., c_tr) to this Grassmannian Schubert data: c_j is the number of indices i_l which equal b_j. Consider the intersection of the Grassmannian Schubert varieties

Z_{(a¹, i₁)} F(t₁), Z_{(a², i₂)} F(t₂), ..., Z_{(a^s, i_s)} F(t_s) ,

(5.12)

where t₁ < t₂ < ... < t_s are in R and F.(t) is the flag osculating the rational normal curve g at g(t).

Conjecture 5.12 Let b = 1 < b₁ < b₂ < ... < b_r < n and (a¹, i₁), (a², i₂), ..., (a^s, i_s) be Grassmannian Schubert data for Fl_b.

If the indices (i₁, i₂, ..., i_s) are in order, then the intersection of the Grassmannian Schubert varieties (5.12) is (a) transverse with (b) with all points of intersection real.
If the indices (i₁, i₂, ..., i_s) are not in order, then there exist numbers t₁ < t₂ < ... < t_s in R for which the intersection of Grassmannian Schubert varieties (5.12) is transverse with all points real and there exist such choices of the t_i such that the intersection is transverse with not all points real.

In particular, enumerative problems involving Grassmannian Schubert varieties on Fl_b are fully real.

Remark 5.13

When each |a|=1, for every possible ordering of the indices (i₁, i₂, ..., i_s) there are real numbers points t₁ < t₂ < ... < t_s such that the intersection (5.12) is transverse with all points real [S08, Corollary 2.2].
When r=1, the flag manifold Fl_b is the Grassmannian Gr(b₁, n) and it is no condition on the indices (i₁, i₂, ..., i_s) to be ordered, so this case of Conjecture 5.12 reduces to Conjecture 5.1.
There is considerable evidence for this conjecture when r=2. Part (2) is true for every set of Grassmannian Schubert data in Fl_2<3C⁵ and all except one such set in Fl_2<4C⁶ (we are presently unable to compute any examples with this Grassmannian Schubert data). Many instances of these same enumerative problems have been computed, and in each instance of (1) the intersection (5.12) is transverse with all points real.
As in the case of Conjecture 5.1, (a) implies (b) in Part (1) of Conjecture 5.12.
We have tested no instances of the intersection (5.12) with r>2, so the truth may differ from the exact statement of Conjecture 5.12.
Conjecture 5.12 has nothing to say when the Schubert data are not Grassmannian.

^§This was in fact its original form. For a short discussion, follow this link.