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Classification of Schubert Galois Groups in \(\textit{Gr}\,(4,9)\)

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Abstract

We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, there are only 149 whose Galois group does not contain the alternating group. We identify the Galois groups of these 149—each is an imprimitive permutation group. These 149 fall into two families according to their geometry. This study suggests a possible classification of Schubert problems whose Galois group is not the full symmetric group, and is a first step toward the inverse Galois problem for Schubert calculus.

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Notes

  1. http://math.stanford.edu/~vakil/programs/galois.

  2. Available at https://www.math.tamu.edu/~sottile/research/stories/GIVIX.

  3. In [31], the term ‘reduced’ is used instead of ‘essential’.

  4. http://math.stanford.edu/~vakil/programs/galois.

  5. https://www.math.tamu.edu/~sottile/research/stories/GIVIX.

  6. https://www.math.tamu.edu/~sottile/research/stories/GIVIX.

  7. https://www.math.tamu.edu/~sottile/research/stories/GIVIX.

  8. https://www.math.tamu.edu/~sottile/research/stories/GIVIX.

  9. https://www.math.tamu.edu/~sottile/research/stories/GIVIX.

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Authors and Affiliations

  1. Centro de Investigación en Matemáticas, A.C., Jalisco S/N, Col. Valenciana, 36023, Guanajuato, Gto., Mexico

    Abraham Martín del Campo

  2. Department of Mathematics, Texas A &M University, College Station, TX, 77843, USA

    Frank Sottile

  3. Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, IN, 47803, USA

    Robert Lee Williams

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  1. Abraham Martín del Campo
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Correspondence to Frank Sottile.

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The work of Martín del Campo was supported in part by CONACyT under Grant Cátedra-1076. The work of Sottile was supported in part by the National Science Foundation under Grant DMS-1501370.

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del Campo, A.M., Sottile, F. & Williams, R.L. Classification of Schubert Galois Groups in \(\textit{Gr}\,(4,9)\). Arnold Math J. 9, 393–433 (2023). https://doi.org/10.1007/s40598-022-00221-2

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