Reference request: Grassmannian and Plucker coordinates in type B, C, D The Next CEO of Stack OverflowInfinite Grassmannians and their coordinate ringsReference request: representation of type G2 Lie algebras.Borel–Weil theorem - reference requestRelations between affine Grassmannian and GrassmannianSubspaces of Grassmannian under Plucker embeddingLattice model for Affine Grassmannians of non type ADo we have super Plucker relations for a super Grassmannian?Reference request: type C, D Catalan numbersReference request: Catalan number of type BDecomposition of product of two Plucker coordinates

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Reference request: Grassmannian and Plucker coordinates in type B, C, D



The Next CEO of Stack OverflowInfinite Grassmannians and their coordinate ringsReference request: representation of type G2 Lie algebras.Borel–Weil theorem - reference requestRelations between affine Grassmannian and GrassmannianSubspaces of Grassmannian under Plucker embeddingLattice model for Affine Grassmannians of non type ADo we have super Plucker relations for a super Grassmannian?Reference request: type C, D Catalan numbersReference request: Catalan number of type BDecomposition of product of two Plucker coordinates










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Grassmannian $Gr(k,n)$ is the set of $k$-dimensional subspace of an $n$-dimensional vector space. What are the Grassmannian in types B, C, D? What are the analog of Plucker coordinates and Plucker relations in these cases? Are there some references about this? Thank you very much.










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    10












    $begingroup$


    Grassmannian $Gr(k,n)$ is the set of $k$-dimensional subspace of an $n$-dimensional vector space. What are the Grassmannian in types B, C, D? What are the analog of Plucker coordinates and Plucker relations in these cases? Are there some references about this? Thank you very much.










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      10












      10








      10


      4



      $begingroup$


      Grassmannian $Gr(k,n)$ is the set of $k$-dimensional subspace of an $n$-dimensional vector space. What are the Grassmannian in types B, C, D? What are the analog of Plucker coordinates and Plucker relations in these cases? Are there some references about this? Thank you very much.










      share|cite|improve this question









      $endgroup$




      Grassmannian $Gr(k,n)$ is the set of $k$-dimensional subspace of an $n$-dimensional vector space. What are the Grassmannian in types B, C, D? What are the analog of Plucker coordinates and Plucker relations in these cases? Are there some references about this? Thank you very much.







      ag.algebraic-geometry co.combinatorics rt.representation-theory lie-groups






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      asked yesterday









      Jianrong LiJianrong Li

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      2,52521319




















          3 Answers
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          11












          $begingroup$

          In type $B$ and $D$ these are orthogonal isotropic Grassmannians $$OGr(k,n) subset Gr(k,n),$$ that parameterize isotropic $k$-dimensional subspaces in a vector space of dimension $n$ endowed with a non-degenerate quadratic form.



          In type $C$ these are symplectic isotropic Grassmannians $$SGr(k,n) subset Gr(k,n),$$ that parameterize isotropic $k$-dimensional subspaces in a vector space of dimension $n$ endowed with a non-degenerate symplectic form.






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            10












            $begingroup$

            What these have in common is that they are of the form $G/P$ for $P$ a maximal parabolic. As such each has a minimal projective embedding of the form $G/P hookrightarrow mathbb P(V_omega)$ where $V_omega$ is a fundamental representation (the analogue of the Plücker coordinates). The two coordinate rings, of $mathbb P(V_omega)$ and $G/P$, are $Sym(V_omega^*)$ and $oplus_n V_nomega^*$ respectively. The kernel of the ring map $Sym(V_omega^*) twoheadrightarrow oplus_n V_nomega^*$ is generated in degree $2$ by Ramanathan's theorem (whose proof you can read in the unique book by Brion + Kumar), i.e. the analogue of the Plücker relations is the complement to $V_2omega^*$ inside $Sym^2(V_omega^*)$. I don't know enough about that representation theory in the specific $BCD$ cases to tell you more.






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              7












              $begingroup$

              It might also be worth mentioning that the Type A Grassmannians are minuscule varieties, meaning they are $G/P$ for a maximal parabolic $P$ corresponding to a minuscule node of the Dynkin diagram. Minuscule (and also cominuscule) varieties tend to behave a bit better than arbitrary homogeneous spaces $G/P$. At least, their combinatorics can be described quite explicitly, like with the Grassmannian: see e.g. https://arxiv.org/abs/math/0608276 or https://arxiv.org/abs/1306.5419. Note that outside of Type A not so many of the nodes of a Dynkin diagram are minuscule.






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                3 Answers
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                active

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                11












                $begingroup$

                In type $B$ and $D$ these are orthogonal isotropic Grassmannians $$OGr(k,n) subset Gr(k,n),$$ that parameterize isotropic $k$-dimensional subspaces in a vector space of dimension $n$ endowed with a non-degenerate quadratic form.



                In type $C$ these are symplectic isotropic Grassmannians $$SGr(k,n) subset Gr(k,n),$$ that parameterize isotropic $k$-dimensional subspaces in a vector space of dimension $n$ endowed with a non-degenerate symplectic form.






                share|cite|improve this answer









                $endgroup$

















                  11












                  $begingroup$

                  In type $B$ and $D$ these are orthogonal isotropic Grassmannians $$OGr(k,n) subset Gr(k,n),$$ that parameterize isotropic $k$-dimensional subspaces in a vector space of dimension $n$ endowed with a non-degenerate quadratic form.



                  In type $C$ these are symplectic isotropic Grassmannians $$SGr(k,n) subset Gr(k,n),$$ that parameterize isotropic $k$-dimensional subspaces in a vector space of dimension $n$ endowed with a non-degenerate symplectic form.






                  share|cite|improve this answer









                  $endgroup$















                    11












                    11








                    11





                    $begingroup$

                    In type $B$ and $D$ these are orthogonal isotropic Grassmannians $$OGr(k,n) subset Gr(k,n),$$ that parameterize isotropic $k$-dimensional subspaces in a vector space of dimension $n$ endowed with a non-degenerate quadratic form.



                    In type $C$ these are symplectic isotropic Grassmannians $$SGr(k,n) subset Gr(k,n),$$ that parameterize isotropic $k$-dimensional subspaces in a vector space of dimension $n$ endowed with a non-degenerate symplectic form.






                    share|cite|improve this answer









                    $endgroup$



                    In type $B$ and $D$ these are orthogonal isotropic Grassmannians $$OGr(k,n) subset Gr(k,n),$$ that parameterize isotropic $k$-dimensional subspaces in a vector space of dimension $n$ endowed with a non-degenerate quadratic form.



                    In type $C$ these are symplectic isotropic Grassmannians $$SGr(k,n) subset Gr(k,n),$$ that parameterize isotropic $k$-dimensional subspaces in a vector space of dimension $n$ endowed with a non-degenerate symplectic form.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered yesterday









                    SashaSasha

                    21.2k22756




                    21.2k22756





















                        10












                        $begingroup$

                        What these have in common is that they are of the form $G/P$ for $P$ a maximal parabolic. As such each has a minimal projective embedding of the form $G/P hookrightarrow mathbb P(V_omega)$ where $V_omega$ is a fundamental representation (the analogue of the Plücker coordinates). The two coordinate rings, of $mathbb P(V_omega)$ and $G/P$, are $Sym(V_omega^*)$ and $oplus_n V_nomega^*$ respectively. The kernel of the ring map $Sym(V_omega^*) twoheadrightarrow oplus_n V_nomega^*$ is generated in degree $2$ by Ramanathan's theorem (whose proof you can read in the unique book by Brion + Kumar), i.e. the analogue of the Plücker relations is the complement to $V_2omega^*$ inside $Sym^2(V_omega^*)$. I don't know enough about that representation theory in the specific $BCD$ cases to tell you more.






                        share|cite|improve this answer









                        $endgroup$

















                          10












                          $begingroup$

                          What these have in common is that they are of the form $G/P$ for $P$ a maximal parabolic. As such each has a minimal projective embedding of the form $G/P hookrightarrow mathbb P(V_omega)$ where $V_omega$ is a fundamental representation (the analogue of the Plücker coordinates). The two coordinate rings, of $mathbb P(V_omega)$ and $G/P$, are $Sym(V_omega^*)$ and $oplus_n V_nomega^*$ respectively. The kernel of the ring map $Sym(V_omega^*) twoheadrightarrow oplus_n V_nomega^*$ is generated in degree $2$ by Ramanathan's theorem (whose proof you can read in the unique book by Brion + Kumar), i.e. the analogue of the Plücker relations is the complement to $V_2omega^*$ inside $Sym^2(V_omega^*)$. I don't know enough about that representation theory in the specific $BCD$ cases to tell you more.






                          share|cite|improve this answer









                          $endgroup$















                            10












                            10








                            10





                            $begingroup$

                            What these have in common is that they are of the form $G/P$ for $P$ a maximal parabolic. As such each has a minimal projective embedding of the form $G/P hookrightarrow mathbb P(V_omega)$ where $V_omega$ is a fundamental representation (the analogue of the Plücker coordinates). The two coordinate rings, of $mathbb P(V_omega)$ and $G/P$, are $Sym(V_omega^*)$ and $oplus_n V_nomega^*$ respectively. The kernel of the ring map $Sym(V_omega^*) twoheadrightarrow oplus_n V_nomega^*$ is generated in degree $2$ by Ramanathan's theorem (whose proof you can read in the unique book by Brion + Kumar), i.e. the analogue of the Plücker relations is the complement to $V_2omega^*$ inside $Sym^2(V_omega^*)$. I don't know enough about that representation theory in the specific $BCD$ cases to tell you more.






                            share|cite|improve this answer









                            $endgroup$



                            What these have in common is that they are of the form $G/P$ for $P$ a maximal parabolic. As such each has a minimal projective embedding of the form $G/P hookrightarrow mathbb P(V_omega)$ where $V_omega$ is a fundamental representation (the analogue of the Plücker coordinates). The two coordinate rings, of $mathbb P(V_omega)$ and $G/P$, are $Sym(V_omega^*)$ and $oplus_n V_nomega^*$ respectively. The kernel of the ring map $Sym(V_omega^*) twoheadrightarrow oplus_n V_nomega^*$ is generated in degree $2$ by Ramanathan's theorem (whose proof you can read in the unique book by Brion + Kumar), i.e. the analogue of the Plücker relations is the complement to $V_2omega^*$ inside $Sym^2(V_omega^*)$. I don't know enough about that representation theory in the specific $BCD$ cases to tell you more.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered yesterday









                            Allen KnutsonAllen Knutson

                            21.4k445129




                            21.4k445129





















                                7












                                $begingroup$

                                It might also be worth mentioning that the Type A Grassmannians are minuscule varieties, meaning they are $G/P$ for a maximal parabolic $P$ corresponding to a minuscule node of the Dynkin diagram. Minuscule (and also cominuscule) varieties tend to behave a bit better than arbitrary homogeneous spaces $G/P$. At least, their combinatorics can be described quite explicitly, like with the Grassmannian: see e.g. https://arxiv.org/abs/math/0608276 or https://arxiv.org/abs/1306.5419. Note that outside of Type A not so many of the nodes of a Dynkin diagram are minuscule.






                                share|cite|improve this answer









                                $endgroup$

















                                  7












                                  $begingroup$

                                  It might also be worth mentioning that the Type A Grassmannians are minuscule varieties, meaning they are $G/P$ for a maximal parabolic $P$ corresponding to a minuscule node of the Dynkin diagram. Minuscule (and also cominuscule) varieties tend to behave a bit better than arbitrary homogeneous spaces $G/P$. At least, their combinatorics can be described quite explicitly, like with the Grassmannian: see e.g. https://arxiv.org/abs/math/0608276 or https://arxiv.org/abs/1306.5419. Note that outside of Type A not so many of the nodes of a Dynkin diagram are minuscule.






                                  share|cite|improve this answer









                                  $endgroup$















                                    7












                                    7








                                    7





                                    $begingroup$

                                    It might also be worth mentioning that the Type A Grassmannians are minuscule varieties, meaning they are $G/P$ for a maximal parabolic $P$ corresponding to a minuscule node of the Dynkin diagram. Minuscule (and also cominuscule) varieties tend to behave a bit better than arbitrary homogeneous spaces $G/P$. At least, their combinatorics can be described quite explicitly, like with the Grassmannian: see e.g. https://arxiv.org/abs/math/0608276 or https://arxiv.org/abs/1306.5419. Note that outside of Type A not so many of the nodes of a Dynkin diagram are minuscule.






                                    share|cite|improve this answer









                                    $endgroup$



                                    It might also be worth mentioning that the Type A Grassmannians are minuscule varieties, meaning they are $G/P$ for a maximal parabolic $P$ corresponding to a minuscule node of the Dynkin diagram. Minuscule (and also cominuscule) varieties tend to behave a bit better than arbitrary homogeneous spaces $G/P$. At least, their combinatorics can be described quite explicitly, like with the Grassmannian: see e.g. https://arxiv.org/abs/math/0608276 or https://arxiv.org/abs/1306.5419. Note that outside of Type A not so many of the nodes of a Dynkin diagram are minuscule.







                                    share|cite|improve this answer












                                    share|cite|improve this answer



                                    share|cite|improve this answer










                                    answered yesterday









                                    Sam HopkinsSam Hopkins

                                    4,95212557




                                    4,95212557



























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