Abstract
We consider tangent cones of Schubert varieties in the complete flag variety, and investigate the problem when the tangent cones of two different Schubert varieties coincide. We give a sufficient condition for such coincidence, and formulate a conjecture that provides a necessary condition. In particular, we show that all Schubert varieties corresponding to the Coxeter elements of the Weyl group have the same tangent cone. Our main tool is the notion of pillar entries in the rank matrix counting the dimensions of the intersections of a given flag with the standard one. This notion is a version of Fulton’s essential set. We calculate the dimension of a Schubert variety in terms of the pillar entries of the rank matrix.
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Notes
We are grateful to M. Kashiwara for a simple proof.
These numbers are obtained using computer programs.
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Acknowledgements
This work was started during our stay at the American Institute of Mathematics within the research program SQuaRE. The final version was completed during a RiP stay at the Mathematisches Forschungsinstitut Oberwolfach. We are grateful to AIM and MFO for their hospitality. We are also grateful to M. Kashiwara and A. Panov for helpful discussions, and to the referee for his/her helpful comments. S. M-G. was partially supported by the ANR project SC\(^{3}\)A, ANR-15-CE40-0004-01.
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Appendix
Appendix
1.1 A1 Comparison of Pillar Entries to Essential Entries
Below are a series of examples and comments about the relationship between pillar entries and Fulton’s essential entries, see also Woo (2009). Recall that essential entries are boxed (while pillar entries are encircled as above).
Let us consider examples that emphasize the difference between the notions of essential and pillar entries. The most interesting case is that of the Coxeter elements.
Example 6.2
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(a)
The rank matrix of the element \(w_0=4\,3\,2\,1\) in \(S_4\) has three essential entries
and no pillar entries. It can be deduced from formula (1), that, for an arbitrary n, the only rank matrix without pillar entries is the matrix \(r(w_0)\) of the longest element \(w_0\in {}S_n\). This matrix has \(n-2\) essential entries along the antidiagonal.
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(b)
For each of the elements \(w_1=2\,1\,4\,3\) and \(w_2=4\,2\,3\,1\) of \(S_4\), we have two essential entries and one pillar:
Note that the position of the pillar entry in the above matrices is the same, while those of the essential entries are different.
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(c)
For the Coxeter elements of \(S_{4}\), we have:
1.2 A2 Rothe Diagrams and Opposite Rothe Diagrams
The Rothe diagram (Rothe 1800) of a permutation \(w\in S_{n}\) is an \(n\times {}n\) square table obtained according to the following rule. Dot the cell (i, j) whenever \(w(i)=j\), shade all the cells of the row at the right of the dotted cell and all the cells of the column below the dotted cell (including the dotted cell). Note that the length \(\ell (w)\) is equal to the number of white cells in the Rothe diagram.
It was noticed in Fulton (1992), that the white cells having a South and East frontier with the shaded region give the positions of the essential entries in the corresponding rank matrix. The value of an essential entry is equal to the number of dots in the upper left quadrant of the Rothe diagram with the origin at the corresponding cell. Let us explain a similar rule to obtain positions of pillar entries.
Consider the opposite Rothe diagram obtained with the following rule. Shade all the cells of the row strictlty at the left of the dotted cell and all the cells of the column strictly above the dotted cell (the dotted cell is not shaded). Note that the number of white undotted cells in the opposite Rothe diagram is equal to \(\ell (w)\) (Table 1).
It follows directly from Definition 2.6, that the white cells having a South and East frontier with the shaded region in the opposite Rothe diagram give the positions of the pillar entries in the corresponding rank matrix. The value of a pillar entry is equal to the number of dots in the upper left quadrant of the diagram.
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Fuchs, D., Kirillov, A., Morier-Genoud, S. et al. On Tangent Cones of Schubert Varieties. Arnold Math J. 3, 451–482 (2017). https://doi.org/10.1007/s40598-017-0074-x
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DOI: https://doi.org/10.1007/s40598-017-0074-x