Abstract
Matrix Schubert varieties are certain varieties in the affine space of square matrices which are determined by specifying rank conditions on submatrices. We study these varieties for generic matrices, symmetric matrices, and upper triangular matrices in view of two applications to algebraic statistics: We observe that special conditional independence models for Gaussian random variables are intersections of matrix Schubert varieties in the symmetric case. Consequently, we obtain a combinatorial primary decomposition algorithm for some conditional independence ideals. We also characterize the vanishing ideals of Gaussian graphical models for generalized Markov chains. In the course of this investigation, we are led to consider three related stratifications, which come from the Schubert stratification of a flag variety. We provide some combinatorial results, including describing the stratifications using the language of rank arrays and enumerating the strata in each case.
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Notes
The upper triangular Gröbner basis result can also be obtained by appealing to [24].
By a diagonal monomial order, we mean any monomial order satisfying the property that the leading term of the determinant of a submatrix is the product of the entries along the diagonal of that submatrix.
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Acknowledgments
We thank Allen Knutson for pointing out the connection between symmetric matrices and the symplectic Grassmannian, and an anonymous referee for several valuable suggestions. In particular, Proposition 4.8 and all connections to Gasharov and Reiner’s work [11] appears because of the referee’s comments. Seth Sullivant was partially supported by the David and Lucille Packard Foundation and the US National Science Foundation (DMS 0954865).
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Fink, A., Rajchgot, J. & Sullivant, S. Matrix Schubert varieties and Gaussian conditional independence models. J Algebr Comb 44, 1009–1046 (2016). https://doi.org/10.1007/s10801-016-0698-2
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DOI: https://doi.org/10.1007/s10801-016-0698-2