Why is the Complete Flag Variety an algebraic variety?












3














Let $V$ be a $mathbb C $ - vector space of dimension $n$.
Let's consider the set $Fl(n)$ of all the complete flags $F_{bullet}$ $$F_1 subset F_2 cdots subset F_n$$ where the $F_i$ are subspaces of $V$ with $dim(F_i)=i$ for every $1 leq i leq n$.



Why is this an affine/projective variety? I know that given that we can use the transitive action of $GL(n, mathbb C)$ and deduce that $$Fl(n) simeq GL(n, mathbb C)/B_n$$ where $B_n$ is the subgroup of the upper triangular matrices. But first we need to show that it is an algebraic variety.



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    3














    Let $V$ be a $mathbb C $ - vector space of dimension $n$.
    Let's consider the set $Fl(n)$ of all the complete flags $F_{bullet}$ $$F_1 subset F_2 cdots subset F_n$$ where the $F_i$ are subspaces of $V$ with $dim(F_i)=i$ for every $1 leq i leq n$.



    Why is this an affine/projective variety? I know that given that we can use the transitive action of $GL(n, mathbb C)$ and deduce that $$Fl(n) simeq GL(n, mathbb C)/B_n$$ where $B_n$ is the subgroup of the upper triangular matrices. But first we need to show that it is an algebraic variety.



    Thanks!










    share|cite|improve this question



























      3












      3








      3







      Let $V$ be a $mathbb C $ - vector space of dimension $n$.
      Let's consider the set $Fl(n)$ of all the complete flags $F_{bullet}$ $$F_1 subset F_2 cdots subset F_n$$ where the $F_i$ are subspaces of $V$ with $dim(F_i)=i$ for every $1 leq i leq n$.



      Why is this an affine/projective variety? I know that given that we can use the transitive action of $GL(n, mathbb C)$ and deduce that $$Fl(n) simeq GL(n, mathbb C)/B_n$$ where $B_n$ is the subgroup of the upper triangular matrices. But first we need to show that it is an algebraic variety.



      Thanks!










      share|cite|improve this question















      Let $V$ be a $mathbb C $ - vector space of dimension $n$.
      Let's consider the set $Fl(n)$ of all the complete flags $F_{bullet}$ $$F_1 subset F_2 cdots subset F_n$$ where the $F_i$ are subspaces of $V$ with $dim(F_i)=i$ for every $1 leq i leq n$.



      Why is this an affine/projective variety? I know that given that we can use the transitive action of $GL(n, mathbb C)$ and deduce that $$Fl(n) simeq GL(n, mathbb C)/B_n$$ where $B_n$ is the subgroup of the upper triangular matrices. But first we need to show that it is an algebraic variety.



      Thanks!







      linear-algebra algebraic-geometry lie-groups schubert-calculus






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      edited yesterday









      Matt Samuel

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      37k63465










      asked Mar 26 '17 at 20:52









      Maffred

      2,670625




      2,670625






















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          because of that identification we can put a variety (projective) structure on it...comes from projective structure of GL(n,C)/B_n.



          there is another identification that this collection of complete flags can be thought of as inside product of G(1,n) x G(2,n) x ... x G(n-1,n)
          where G(r,n) Grassmanian variety .






          share|cite|improve this answer





















          • I like the second identification! Why is it an algebraic subset of that product?
            – Maffred
            Mar 26 '17 at 21:09






          • 2




            Because you can encode the containment of subspaces as polynomial equations in the Plucker coordinates of the subspaces.
            – Mariano Suárez-Álvarez
            Mar 26 '17 at 21:45











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          1 Answer
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          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          3














          because of that identification we can put a variety (projective) structure on it...comes from projective structure of GL(n,C)/B_n.



          there is another identification that this collection of complete flags can be thought of as inside product of G(1,n) x G(2,n) x ... x G(n-1,n)
          where G(r,n) Grassmanian variety .






          share|cite|improve this answer





















          • I like the second identification! Why is it an algebraic subset of that product?
            – Maffred
            Mar 26 '17 at 21:09






          • 2




            Because you can encode the containment of subspaces as polynomial equations in the Plucker coordinates of the subspaces.
            – Mariano Suárez-Álvarez
            Mar 26 '17 at 21:45
















          3














          because of that identification we can put a variety (projective) structure on it...comes from projective structure of GL(n,C)/B_n.



          there is another identification that this collection of complete flags can be thought of as inside product of G(1,n) x G(2,n) x ... x G(n-1,n)
          where G(r,n) Grassmanian variety .






          share|cite|improve this answer





















          • I like the second identification! Why is it an algebraic subset of that product?
            – Maffred
            Mar 26 '17 at 21:09






          • 2




            Because you can encode the containment of subspaces as polynomial equations in the Plucker coordinates of the subspaces.
            – Mariano Suárez-Álvarez
            Mar 26 '17 at 21:45














          3












          3








          3






          because of that identification we can put a variety (projective) structure on it...comes from projective structure of GL(n,C)/B_n.



          there is another identification that this collection of complete flags can be thought of as inside product of G(1,n) x G(2,n) x ... x G(n-1,n)
          where G(r,n) Grassmanian variety .






          share|cite|improve this answer












          because of that identification we can put a variety (projective) structure on it...comes from projective structure of GL(n,C)/B_n.



          there is another identification that this collection of complete flags can be thought of as inside product of G(1,n) x G(2,n) x ... x G(n-1,n)
          where G(r,n) Grassmanian variety .







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Mar 26 '17 at 21:02









          Pinaki Saha

          312




          312












          • I like the second identification! Why is it an algebraic subset of that product?
            – Maffred
            Mar 26 '17 at 21:09






          • 2




            Because you can encode the containment of subspaces as polynomial equations in the Plucker coordinates of the subspaces.
            – Mariano Suárez-Álvarez
            Mar 26 '17 at 21:45


















          • I like the second identification! Why is it an algebraic subset of that product?
            – Maffred
            Mar 26 '17 at 21:09






          • 2




            Because you can encode the containment of subspaces as polynomial equations in the Plucker coordinates of the subspaces.
            – Mariano Suárez-Álvarez
            Mar 26 '17 at 21:45
















          I like the second identification! Why is it an algebraic subset of that product?
          – Maffred
          Mar 26 '17 at 21:09




          I like the second identification! Why is it an algebraic subset of that product?
          – Maffred
          Mar 26 '17 at 21:09




          2




          2




          Because you can encode the containment of subspaces as polynomial equations in the Plucker coordinates of the subspaces.
          – Mariano Suárez-Álvarez
          Mar 26 '17 at 21:45




          Because you can encode the containment of subspaces as polynomial equations in the Plucker coordinates of the subspaces.
          – Mariano Suárez-Álvarez
          Mar 26 '17 at 21:45


















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