Consider the Schubert variety $X(s_3s_2s_1s_4s_3s_2)$ in $SL_5/P_2$, where $P_2$ is the maximal parabolic corresponding to the simple root $\alpha_2$. In one line notation this permutation can be written as $(45123)$. Since $(4512) > (3412)$ this Schubert variety is Singular by the pattern avoidance criterion. But this Schubert variety is the full Grassmannian $SL_5/P_2$ and hence smooth. Am I missing something ? On the other hand the Schubert variety $X(s_2s_1s_4s_3s_2)$ is singular. What are the irreducible components of the singular locus ?
2026-02-26 10:49:55.1772102995
Smoothness of Schubert Variety
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