Schubert variety associated to a flag of subspaces of a vector space
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
edited yesterday
Matt Samuel
37k63465
asked Feb 26 at 11:17
YoYo
55739
|
show 2 more comments
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
edited yesterday
Matt Samuel
37k63465
asked Feb 26 at 11:17
YoYo
55739
1
Is it clear the definition of $W$? I remember you that $underline{j}notlequnderline{i},,d_{underline{j}}=0$, in my notation!
– Armando j18eos
Mar 27 at 22:24
$ 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.
– YoYo
Mar 28 at 15:18
1
Because lemma 1.4.5 by Littelmann's lecture notes holds.
– Armando j18eos
Mar 28 at 17:12
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. :-)
– YoYo
Mar 28 at 18:32
1
Yes, it is all right. Well done. ;)
– Armando j18eos
Mar 29 at 8:33
|
show 2 more comments
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
edited yesterday
Matt Samuel
37k63465
asked Feb 26 at 11:17
YoYo
55739
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
edited yesterday
Matt Samuel
37k63465
asked Feb 26 at 11:17
YoYo
55739
edited yesterday
Matt Samuel
37k63465
asked Feb 26 at 11:17
YoYo
55739
edited yesterday
Matt Samuel
37k63465
edited yesterday
Matt Samuel
37k63465
edited yesterday
Matt Samuel
37k63465
37k63465
asked Feb 26 at 11:17
YoYo
55739
asked Feb 26 at 11:17
YoYo
55739
asked Feb 26 at 11:17
YoYo
55739
55739
1
Is it clear the definition of $W$? I remember you that $underline{j}notlequnderline{i},,d_{underline{j}}=0$, in my notation!
– Armando j18eos
Mar 27 at 22:24
$ 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.
– YoYo
Mar 28 at 15:18
1
Because lemma 1.4.5 by Littelmann's lecture notes holds.
– Armando j18eos
Mar 28 at 17:12
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. :-)
– YoYo
Mar 28 at 18:32
1
Yes, it is all right. Well done. ;)
– Armando j18eos
Mar 29 at 8:33
|
show 2 more comments
1
Is it clear the definition of $W$? I remember you that $underline{j}notlequnderline{i},,d_{underline{j}}=0$, in my notation!
– Armando j18eos
Mar 27 at 22:24
$ 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.
– YoYo
Mar 28 at 15:18
1
Because lemma 1.4.5 by Littelmann's lecture notes holds.
– Armando j18eos
Mar 28 at 17:12
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. :-)
– YoYo
Mar 28 at 18:32
1
Yes, it is all right. Well done. ;)
– Armando j18eos
Mar 29 at 8:33
1
1
Is it clear the definition of $W$? I remember you that $underline{j}notlequnderline{i},,d_{underline{j}}=0$, in my notation!
– Armando j18eos
Mar 27 at 22:24
Is it clear the definition of $W$? I remember you that $underline{j}notlequnderline{i},,d_{underline{j}}=0$, in my notation!
– Armando j18eos
Mar 27 at 22:24
$ 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.
– YoYo
Mar 28 at 15:18
$ 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.
– YoYo
Mar 28 at 15:18
1
1
Because lemma 1.4.5 by Littelmann's lecture notes holds.
– Armando j18eos
Mar 28 at 17:12
Because lemma 1.4.5 by Littelmann's lecture notes holds.
– Armando j18eos
Mar 28 at 17:12
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. :-)
– YoYo
Mar 28 at 18:32
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. :-)
– YoYo
Mar 28 at 18:32
1
1
Yes, it is all right. Well done. ;)
– Armando j18eos
Mar 29 at 8:33
Yes, it is all right. Well done. ;)
– Armando j18eos
Mar 29 at 8:33
|
show 2 more comments
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1
Is it clear the definition of $W$? I remember you that $underline{j}notlequnderline{i},,d_{underline{j}}=0$, in my notation!
– Armando j18eos
Mar 27 at 22:24
$ 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.
– YoYo
Mar 28 at 15:18
1
Because lemma 1.4.5 by Littelmann's lecture notes holds.
– Armando j18eos
Mar 28 at 17:12
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. :-)
– YoYo
Mar 28 at 18:32
1
Yes, it is all right. Well done. ;)
– Armando j18eos
Mar 29 at 8:33