I am trying to compute the singular locus of the schubert variety $X_w$ in $G_{2,7}$ where $w=(4,7) \in I_{2,7}$. Following the notation in the book "The Grassmannian Variety: Geometric and Representation theoritic aspect" page 94 (also can be found in "Singular Loci in Schubert varieties" by Lakshmibai and Beilly page-138) we have $\lambda =(5,3)$ and $\alpha_1=(1,4)$. Transforming $\alpha_1$ as an element of $G_{2,7}$ we get $(3,5) \in I_{2,7}$. So $X_w$ has one component namely $X_{(3,5)}$. My questions are the following:
$\alpha_1$ is not a partition as the book claims or do we read it as $(4,1)$ instead of reading it as $(1,4)$ ?
What is the hook are we deleting from the young diagram of $\lambda$ ?
What is the singular component in terms of the simple reflection ?