Galois groups of Schubert problems via homotopy computation. (English) Zbl 1210.14064
Let \(G(k,n)\) be the Grassmannian of \(k\)-planes in \(\mathbb C^n\). A Schubert problem in \(G(k,n)\) asks to determine the (number of) \(k\)-planes satisying some fixed incidence conditions, with respect to general fixed flags of subspaces of \(\mathbb C^n\), such that this number is finite. Every incidence condition defines a Schubert variety, therefore the solution of a Schubert problem can be represented by a transverse intersection of Schubert varieties. In particular, by definition, a simple Schubert problem involves only two Schubert conditions of codimension higher than one, it enjoys the property of being a complete intersection.
To a given Schubert problem one naturally associates a Galois group; if it is the full symmetric group, it means that the problem has no underlying structures. The article under review tackles the problem of finding the Galois group of some simple Schubert problems, using the method of homotopy computation, recently developed in the new field of numerical algebraic geometry by Sommese and Wampler (see for instance [A. J. Sommese and C. W. Wampler, II, The numerical solution of systems of polynomials. Arising in engineering and science. River Edge, NJ: World Scientific. (2005; Zbl 1091.65049)]). This method, initially conceived for applications of mathematics, has proved here to be very efficient also to solve purely theoretic problems.
As an example, the following numerical theorem is stated: The Galois group of the Schubert problem of \(3\)-planes in \(\mathbb C^8\) meeting \(15\) fixed \(5\)-planes non-trivially is the full symmetric group on \(6006\) letters.
This kind of problems are intractable with the usual symbolic methods, so it appears that in the next future the numerical algorithms will be further developed. The article includes a precise description of the software used. The lines of future research in this field are also indicated.
The article is clearly written and very pleasant to read.
To a given Schubert problem one naturally associates a Galois group; if it is the full symmetric group, it means that the problem has no underlying structures. The article under review tackles the problem of finding the Galois group of some simple Schubert problems, using the method of homotopy computation, recently developed in the new field of numerical algebraic geometry by Sommese and Wampler (see for instance [A. J. Sommese and C. W. Wampler, II, The numerical solution of systems of polynomials. Arising in engineering and science. River Edge, NJ: World Scientific. (2005; Zbl 1091.65049)]). This method, initially conceived for applications of mathematics, has proved here to be very efficient also to solve purely theoretic problems.
As an example, the following numerical theorem is stated: The Galois group of the Schubert problem of \(3\)-planes in \(\mathbb C^8\) meeting \(15\) fixed \(5\)-planes non-trivially is the full symmetric group on \(6006\) letters.
This kind of problems are intractable with the usual symbolic methods, so it appears that in the next future the numerical algorithms will be further developed. The article includes a precise description of the software used. The lines of future research in this field are also indicated.
The article is clearly written and very pleasant to read.
Reviewer: Emilia Mezzetti (Trieste)
MSC:
| 14N15 | Classical problems, Schubert calculus |
| 65H20 | Global methods, including homotopy approaches to the numerical solution of nonlinear equations |
Citations:
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