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Lower bounds for numbers of real solutions in problems of Schubert calculus. (English) Zbl 1365.14070

It is well known that the problem of finding the number of real solutions to algebraic systems is very difficult, and not many results are known. In this paper the authors address the counting of real points in intersections of Schubert varieties associated with osculating flags in the Grassmannian of \(n\)-dimensional planes in a \(d\)-dimensional space. These problems are parameterized by partitions \(\lambda^{(1)}, \dots, \lambda^{(k)}\) and \(\nu\) with at most \(n\) parts satisfying the condition \(|\nu|+ \sum_{i=1}^k|\lambda^{(i)}| =n(d-n)\), and distinct complex numbers \(z_1, \dots, z_k\).
In this parametrization,\(\lambda^{(1)}, \dots, \lambda^{(k)}\) and \(\nu\) are respectively paired with \(z_1, \dots, z_k\) and infinity.
Equivalently, they count \(n\)-dimensional real vector spaces of polynomials that have ramification points \(z_1, \dots, z_k\) with respective ramification conditions \(\lambda^{(1)}, \dots, \lambda^{(k)}\) and are spanned by polynomials of degrees \(d-i-\nu_{n+1-i},i=1,\dots, n\).
The same number is obtained by counting real monic monodromy-free Fuchsian differential operators with singular points \(z_1, \dots, z_k\) and infinity, exponents \({\lambda_{n}}^{(i)}, {\lambda_{n-1}}^{(i)} + 1,\dots, {\lambda_{1}}^{(i)}+ n-1\) at the \(z_i\)’s, \(i=1,\dots, k\), and exponents \(\nu_n + 1 - d,\nu_{n-1} + 2 - d, \dots,\nu_{1} + n - d \) at infinity.
The number of complex solutions to the above-mentioned algebraic systems is readily given by the Schubert calculus and equals the multiplicity of the irreducible Verma module \(L_{\mu}\) with highest weight \(\mu\).
Reviewer: Cenap Özel (Bolu)

MSC:

14N15 Classical problems, Schubert calculus
14M15 Grassmannians, Schubert varieties, flag manifolds
14P05 Real algebraic sets
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