Schubert variety associated to a flag of subspaces of a vector space












0












$begingroup$


At the end of the following page : The associated Schubert variety of a flag of subspaces of a vector space. , the author says :



Let $[W]in X=X_{underline i}$, by construction:
begin{equation}
[W]=[v_1wedgedotswedge v_k]=left[sum_{underline{j}lequnderline{i}}d_{underline j}e_{underline j}right]
end{equation}
in particular:
begin{equation}
forall hin{1,dots,k},,dimleft(Wcap F_{i_h}right)geq h
end{equation}
and vice versa.



Could someone explain to me why have we this, in details please ?



Thanks in advance for your help.






algebraic-geometry algebraic-groups grassmannian schubert-calculus







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Is it clear the definition of $W$? I remember you that $underline{j}notlequnderline{i},,d_{underline{j}}=0$, in my notation!
    $endgroup$
    – Armando j18eos
    Mar 27 '18 at 22:24










  • $begingroup$
    $ W in X_{ underline{i} } = { W in mathrm{Gr}_{d,n} | d_{ underline{j} } = 0 , forall j in I_{d,n} text{such that} underline{j} not leq underline{i} } $, but, why is $ X_{ underline{i} } = { W in mathrm{Gr}_{d,n} | dim (W cap F_{i_{h}} geq h } $ ? Thank you.
    $endgroup$
    – YoYo
    Mar 28 '18 at 15:18






  • 1




    $begingroup$
    Because lemma 1.4.5 by Littelmann's lecture notes holds.
    $endgroup$
    – Armando j18eos
    Mar 28 '18 at 17:12












  • $begingroup$
    Yes, i know, but untill now, i'm not able to make a link between lemma 1.4.5 and : $ X_{ underline{i} } = { W in mathrm{Gr_{d,n}} | dim ( W cap F_{i_{h}} ) geq h } $. Could you try to explain to me this point please ? Thank you. :-)
    $endgroup$
    – YoYo
    Mar 28 '18 at 18:32








  • 1




    $begingroup$
    Yes, it is all right. Well done. ;)
    $endgroup$
    – Armando j18eos
    Mar 29 '18 at 8:33
















0












$begingroup$


At the end of the following page : The associated Schubert variety of a flag of subspaces of a vector space. , the author says :



Let $[W]in X=X_{underline i}$, by construction:
begin{equation}
[W]=[v_1wedgedotswedge v_k]=left[sum_{underline{j}lequnderline{i}}d_{underline j}e_{underline j}right]
end{equation}
in particular:
begin{equation}
forall hin{1,dots,k},,dimleft(Wcap F_{i_h}right)geq h
end{equation}
and vice versa.



Could someone explain to me why have we this, in details please ?



Thanks in advance for your help.






algebraic-geometry algebraic-groups grassmannian schubert-calculus







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Is it clear the definition of $W$? I remember you that $underline{j}notlequnderline{i},,d_{underline{j}}=0$, in my notation!
    $endgroup$
    – Armando j18eos
    Mar 27 '18 at 22:24










  • $begingroup$
    $ W in X_{ underline{i} } = { W in mathrm{Gr}_{d,n} | d_{ underline{j} } = 0 , forall j in I_{d,n} text{such that} underline{j} not leq underline{i} } $, but, why is $ X_{ underline{i} } = { W in mathrm{Gr}_{d,n} | dim (W cap F_{i_{h}} geq h } $ ? Thank you.
    $endgroup$
    – YoYo
    Mar 28 '18 at 15:18






  • 1




    $begingroup$
    Because lemma 1.4.5 by Littelmann's lecture notes holds.
    $endgroup$
    – Armando j18eos
    Mar 28 '18 at 17:12












  • $begingroup$
    Yes, i know, but untill now, i'm not able to make a link between lemma 1.4.5 and : $ X_{ underline{i} } = { W in mathrm{Gr_{d,n}} | dim ( W cap F_{i_{h}} ) geq h } $. Could you try to explain to me this point please ? Thank you. :-)
    $endgroup$
    – YoYo
    Mar 28 '18 at 18:32








  • 1




    $begingroup$
    Yes, it is all right. Well done. ;)
    $endgroup$
    – Armando j18eos
    Mar 29 '18 at 8:33














0












0








0





$begingroup$


At the end of the following page : The associated Schubert variety of a flag of subspaces of a vector space. , the author says :



Let $[W]in X=X_{underline i}$, by construction:
begin{equation}
[W]=[v_1wedgedotswedge v_k]=left[sum_{underline{j}lequnderline{i}}d_{underline j}e_{underline j}right]
end{equation}
in particular:
begin{equation}
forall hin{1,dots,k},,dimleft(Wcap F_{i_h}right)geq h
end{equation}
and vice versa.



Could someone explain to me why have we this, in details please ?



Thanks in advance for your help.






algebraic-geometry algebraic-groups grassmannian schubert-calculus







share|cite|improve this question











$endgroup$




At the end of the following page : The associated Schubert variety of a flag of subspaces of a vector space. , the author says :



Let $[W]in X=X_{underline i}$, by construction:
begin{equation}
[W]=[v_1wedgedotswedge v_k]=left[sum_{underline{j}lequnderline{i}}d_{underline j}e_{underline j}right]
end{equation}
in particular:
begin{equation}
forall hin{1,dots,k},,dimleft(Wcap F_{i_h}right)geq h
end{equation}
and vice versa.



Could someone explain to me why have we this, in details please ?



Thanks in advance for your help.






algebraic-geometry algebraic-groups grassmannian schubert-calculus




algebraic-geometry algebraic-groups grassmannian schubert-calculus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 25 '18 at 11:47









Matt Samuel

39.2k63770




39.2k63770










asked Feb 26 '18 at 11:17









YoYoYoYo

56739




56739








  • 1




    $begingroup$
    Is it clear the definition of $W$? I remember you that $underline{j}notlequnderline{i},,d_{underline{j}}=0$, in my notation!
    $endgroup$
    – Armando j18eos
    Mar 27 '18 at 22:24










  • $begingroup$
    $ W in X_{ underline{i} } = { W in mathrm{Gr}_{d,n} | d_{ underline{j} } = 0 , forall j in I_{d,n} text{such that} underline{j} not leq underline{i} } $, but, why is $ X_{ underline{i} } = { W in mathrm{Gr}_{d,n} | dim (W cap F_{i_{h}} geq h } $ ? Thank you.
    $endgroup$
    – YoYo
    Mar 28 '18 at 15:18






  • 1




    $begingroup$
    Because lemma 1.4.5 by Littelmann's lecture notes holds.
    $endgroup$
    – Armando j18eos
    Mar 28 '18 at 17:12












  • $begingroup$
    Yes, i know, but untill now, i'm not able to make a link between lemma 1.4.5 and : $ X_{ underline{i} } = { W in mathrm{Gr_{d,n}} | dim ( W cap F_{i_{h}} ) geq h } $. Could you try to explain to me this point please ? Thank you. :-)
    $endgroup$
    – YoYo
    Mar 28 '18 at 18:32








  • 1




    $begingroup$
    Yes, it is all right. Well done. ;)
    $endgroup$
    – Armando j18eos
    Mar 29 '18 at 8:33














  • 1




    $begingroup$
    Is it clear the definition of $W$? I remember you that $underline{j}notlequnderline{i},,d_{underline{j}}=0$, in my notation!
    $endgroup$
    – Armando j18eos
    Mar 27 '18 at 22:24










  • $begingroup$
    $ W in X_{ underline{i} } = { W in mathrm{Gr}_{d,n} | d_{ underline{j} } = 0 , forall j in I_{d,n} text{such that} underline{j} not leq underline{i} } $, but, why is $ X_{ underline{i} } = { W in mathrm{Gr}_{d,n} | dim (W cap F_{i_{h}} geq h } $ ? Thank you.
    $endgroup$
    – YoYo
    Mar 28 '18 at 15:18






  • 1




    $begingroup$
    Because lemma 1.4.5 by Littelmann's lecture notes holds.
    $endgroup$
    – Armando j18eos
    Mar 28 '18 at 17:12












  • $begingroup$
    Yes, i know, but untill now, i'm not able to make a link between lemma 1.4.5 and : $ X_{ underline{i} } = { W in mathrm{Gr_{d,n}} | dim ( W cap F_{i_{h}} ) geq h } $. Could you try to explain to me this point please ? Thank you. :-)
    $endgroup$
    – YoYo
    Mar 28 '18 at 18:32








  • 1




    $begingroup$
    Yes, it is all right. Well done. ;)
    $endgroup$
    – Armando j18eos
    Mar 29 '18 at 8:33








1




1




$begingroup$
Is it clear the definition of $W$? I remember you that $underline{j}notlequnderline{i},,d_{underline{j}}=0$, in my notation!
$endgroup$
– Armando j18eos
Mar 27 '18 at 22:24




$begingroup$
Is it clear the definition of $W$? I remember you that $underline{j}notlequnderline{i},,d_{underline{j}}=0$, in my notation!
$endgroup$
– Armando j18eos
Mar 27 '18 at 22:24












$begingroup$
$ W in X_{ underline{i} } = { W in mathrm{Gr}_{d,n} | d_{ underline{j} } = 0 , forall j in I_{d,n} text{such that} underline{j} not leq underline{i} } $, but, why is $ X_{ underline{i} } = { W in mathrm{Gr}_{d,n} | dim (W cap F_{i_{h}} geq h } $ ? Thank you.
$endgroup$
– YoYo
Mar 28 '18 at 15:18




$begingroup$
$ W in X_{ underline{i} } = { W in mathrm{Gr}_{d,n} | d_{ underline{j} } = 0 , forall j in I_{d,n} text{such that} underline{j} not leq underline{i} } $, but, why is $ X_{ underline{i} } = { W in mathrm{Gr}_{d,n} | dim (W cap F_{i_{h}} geq h } $ ? Thank you.
$endgroup$
– YoYo
Mar 28 '18 at 15:18




1




1




$begingroup$
Because lemma 1.4.5 by Littelmann's lecture notes holds.
$endgroup$
– Armando j18eos
Mar 28 '18 at 17:12






$begingroup$
Because lemma 1.4.5 by Littelmann's lecture notes holds.
$endgroup$
– Armando j18eos
Mar 28 '18 at 17:12














$begingroup$
Yes, i know, but untill now, i'm not able to make a link between lemma 1.4.5 and : $ X_{ underline{i} } = { W in mathrm{Gr_{d,n}} | dim ( W cap F_{i_{h}} ) geq h } $. Could you try to explain to me this point please ? Thank you. :-)
$endgroup$
– YoYo
Mar 28 '18 at 18:32






$begingroup$
Yes, i know, but untill now, i'm not able to make a link between lemma 1.4.5 and : $ X_{ underline{i} } = { W in mathrm{Gr_{d,n}} | dim ( W cap F_{i_{h}} ) geq h } $. Could you try to explain to me this point please ? Thank you. :-)
$endgroup$
– YoYo
Mar 28 '18 at 18:32






1




1




$begingroup$
Yes, it is all right. Well done. ;)
$endgroup$
– Armando j18eos
Mar 29 '18 at 8:33




$begingroup$
Yes, it is all right. Well done. ;)
$endgroup$
– Armando j18eos
Mar 29 '18 at 8:33










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