Singular Locus of a Schubert variety
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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 ?
combinatorics algebraic-geometry representation-theory schubert-calculus
edited Dec 25 '18 at 16:19
Matt Samuel
38.5k63768
asked Jul 11 '16 at 14:26
JackJack
875
$endgroup$
add a comment |
$begingroup$
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 ?
combinatorics algebraic-geometry representation-theory schubert-calculus
edited Dec 25 '18 at 16:19
Matt Samuel
38.5k63768
asked Jul 11 '16 at 14:26
JackJack
875
$endgroup$
add a comment |
$begingroup$
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 ?
combinatorics algebraic-geometry representation-theory schubert-calculus
edited Dec 25 '18 at 16:19
Matt Samuel
38.5k63768
asked Jul 11 '16 at 14:26
JackJack
875
$endgroup$
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 ?
combinatorics algebraic-geometry representation-theory schubert-calculus
combinatorics algebraic-geometry representation-theory schubert-calculus
edited Dec 25 '18 at 16:19
Matt Samuel
38.5k63768
asked Jul 11 '16 at 14:26
JackJack
875
edited Dec 25 '18 at 16:19
Matt Samuel
38.5k63768
asked Jul 11 '16 at 14:26
JackJack
875
edited Dec 25 '18 at 16:19
Matt Samuel
38.5k63768
edited Dec 25 '18 at 16:19
Matt Samuel
38.5k63768
edited Dec 25 '18 at 16:19
Matt Samuel
38.5k63768
38.5k63768
asked Jul 11 '16 at 14:26
JackJack
875
asked Jul 11 '16 at 14:26
JackJack
875
asked Jul 11 '16 at 14:26
JackJack
875
875
add a comment |
add a comment |
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