If $omega in bigwedge^kmathbb{R}^d$ is decomposable over $mathbb C$, is it decomposable over $mathbb R$?












2












$begingroup$


Let $k,d$ be positive integers, and let $omega in bigwedge^kmathbb{R}^d$ be decomposable in $ bigwedge^kmathbb{C}^d$.



Is $omega$ decomposable in $bigwedge^kmathbb{R}^d$?



Edit:



Let me be more careful about the formulation of this question, as $bigwedge^kmathbb{C}^d$ can have two different non-isomorphic interpretation:





  1. $bigwedge^k_{mathbb{R}}mathbb{C}^d$: Here we take exterior power over $mathbb R$. In particular, we think of $mathbb{C}^d$ as a real $2d$-dimensional vector space. In that case $mathbb{R}^d$ is a vector subspace (over $mathbb{R}$), and so we have the general claim that “being decomposable” is a property which remains invariant under passing to a subspace. So, in that case, the answer is positive.


Sasha's answer refers to this interpretation of the question.





  1. $bigwedge^k_{mathbb{C}}mathbb{C}^d$: Here we take exterior power over $mathbb C$, and we think of $mathbb{C}^d$ as a complex vector space. In this case $mathbb{R}^d$ is not a (complex) vector subspace of $mathbb{C}^d$, but we can still view $bigwedge^kmathbb{R}^d$ as a subspace of $bigwedge^k_{mathbb{C}}mathbb{C}^d$, via complexification. This because complexification commutes with exterior powers, so $(bigwedge^kmathbb{R}^d)^{mathbb C}=bigwedge^k_{mathbb{C}}((mathbb{R}^d)^{mathbb C})=bigwedge^k_{mathbb{C}}mathbb{C}^d.$ Any real vector space $V$ can be viewed as a subspace of its complexification $V^{mathbb C}=V otimes_{mathbb R}C$ via the map $v to v otimes 1$. Thus, we can consider in this way $bigwedge^kmathbb{R}^d$ as a subspace of $(bigwedge^kmathbb{R}^d)^{mathbb C}=bigwedge^k_{mathbb{C}}mathbb{C}^d.$
    Now we are given an element $omega in bigwedge^kmathbb{R}^d$, which is decomposable as an element in $bigwedge^k_{mathbb{C}}mathbb{C}^d$, and we ask whether or not it is decomposable as an element in $ bigwedge^kmathbb{R}^d$.



What is the answer for this second variant of the question?




Comment:



Perhaps there is a way to view $bigwedge^k_{mathbb{C}}mathbb{C}^d$ as a subspace of $bigwedge^k_{mathbb{R}}mathbb{C}^d$ in a way which preserves decomposability, and the "real copy" $bigwedge^k_{mathbb{R}}mathbb{R}^d$, thus reducing the second problem to the first one. I asked about the possible existence of such an embedding here.



So far, I only know that the answer is positive for $k=2$:



Let $omega in bigwedge^2mathbb{R}^d$ be decomposable in $bigwedge^2mathbb{C}^d$. $omega$ can be written as



$$omega=(u_1+iv_1) wedge (u_2+iv_2), tag{1}$$ where $u_1,u_2,v_1,v_2 in mathbb R^d$. Since



$$ omega=(u_1 wedge u_2 - v_1 wedge v_2)+i (v_1 wedge u_2+u_1 wedge v_2), $$



$omega in bigwedge^2mathbb{R}^d$ if and only if $$v_1 wedge u_2=-u_1 wedge v_2, tag{2}$$



where this is an equality in $bigwedge^2mathbb{R}^d$.



Suppose that $omega in bigwedge^2mathbb{R}^d$. If $dim(text{span}_{mathbb R}(u_1,u_2) cap text{span}_{mathbb R}(v_1,v_2)) ge 1$ then $omega=u_1 wedge u_2 - v_1 wedge v_2$ is decomposable in $bigwedge^2mathbb{R}^d$. Otherwise, $u_1,u_2,v_1,v_2$ are linearly independent over $mathbb R$, which violates equation $(2)$.



I don't see an immediate generalization of this proof for $k ge 3$. By expanding $$omega=(u_1+iv_1) wedge (u_2+iv_2) wedge dots wedge (u_k+iv_k)$$
we get more than two summands in the real part.










share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    Let $k,d$ be positive integers, and let $omega in bigwedge^kmathbb{R}^d$ be decomposable in $ bigwedge^kmathbb{C}^d$.



    Is $omega$ decomposable in $bigwedge^kmathbb{R}^d$?



    Edit:



    Let me be more careful about the formulation of this question, as $bigwedge^kmathbb{C}^d$ can have two different non-isomorphic interpretation:





    1. $bigwedge^k_{mathbb{R}}mathbb{C}^d$: Here we take exterior power over $mathbb R$. In particular, we think of $mathbb{C}^d$ as a real $2d$-dimensional vector space. In that case $mathbb{R}^d$ is a vector subspace (over $mathbb{R}$), and so we have the general claim that “being decomposable” is a property which remains invariant under passing to a subspace. So, in that case, the answer is positive.


    Sasha's answer refers to this interpretation of the question.





    1. $bigwedge^k_{mathbb{C}}mathbb{C}^d$: Here we take exterior power over $mathbb C$, and we think of $mathbb{C}^d$ as a complex vector space. In this case $mathbb{R}^d$ is not a (complex) vector subspace of $mathbb{C}^d$, but we can still view $bigwedge^kmathbb{R}^d$ as a subspace of $bigwedge^k_{mathbb{C}}mathbb{C}^d$, via complexification. This because complexification commutes with exterior powers, so $(bigwedge^kmathbb{R}^d)^{mathbb C}=bigwedge^k_{mathbb{C}}((mathbb{R}^d)^{mathbb C})=bigwedge^k_{mathbb{C}}mathbb{C}^d.$ Any real vector space $V$ can be viewed as a subspace of its complexification $V^{mathbb C}=V otimes_{mathbb R}C$ via the map $v to v otimes 1$. Thus, we can consider in this way $bigwedge^kmathbb{R}^d$ as a subspace of $(bigwedge^kmathbb{R}^d)^{mathbb C}=bigwedge^k_{mathbb{C}}mathbb{C}^d.$
      Now we are given an element $omega in bigwedge^kmathbb{R}^d$, which is decomposable as an element in $bigwedge^k_{mathbb{C}}mathbb{C}^d$, and we ask whether or not it is decomposable as an element in $ bigwedge^kmathbb{R}^d$.



    What is the answer for this second variant of the question?




    Comment:



    Perhaps there is a way to view $bigwedge^k_{mathbb{C}}mathbb{C}^d$ as a subspace of $bigwedge^k_{mathbb{R}}mathbb{C}^d$ in a way which preserves decomposability, and the "real copy" $bigwedge^k_{mathbb{R}}mathbb{R}^d$, thus reducing the second problem to the first one. I asked about the possible existence of such an embedding here.



    So far, I only know that the answer is positive for $k=2$:



    Let $omega in bigwedge^2mathbb{R}^d$ be decomposable in $bigwedge^2mathbb{C}^d$. $omega$ can be written as



    $$omega=(u_1+iv_1) wedge (u_2+iv_2), tag{1}$$ where $u_1,u_2,v_1,v_2 in mathbb R^d$. Since



    $$ omega=(u_1 wedge u_2 - v_1 wedge v_2)+i (v_1 wedge u_2+u_1 wedge v_2), $$



    $omega in bigwedge^2mathbb{R}^d$ if and only if $$v_1 wedge u_2=-u_1 wedge v_2, tag{2}$$



    where this is an equality in $bigwedge^2mathbb{R}^d$.



    Suppose that $omega in bigwedge^2mathbb{R}^d$. If $dim(text{span}_{mathbb R}(u_1,u_2) cap text{span}_{mathbb R}(v_1,v_2)) ge 1$ then $omega=u_1 wedge u_2 - v_1 wedge v_2$ is decomposable in $bigwedge^2mathbb{R}^d$. Otherwise, $u_1,u_2,v_1,v_2$ are linearly independent over $mathbb R$, which violates equation $(2)$.



    I don't see an immediate generalization of this proof for $k ge 3$. By expanding $$omega=(u_1+iv_1) wedge (u_2+iv_2) wedge dots wedge (u_k+iv_k)$$
    we get more than two summands in the real part.










    share|cite|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      Let $k,d$ be positive integers, and let $omega in bigwedge^kmathbb{R}^d$ be decomposable in $ bigwedge^kmathbb{C}^d$.



      Is $omega$ decomposable in $bigwedge^kmathbb{R}^d$?



      Edit:



      Let me be more careful about the formulation of this question, as $bigwedge^kmathbb{C}^d$ can have two different non-isomorphic interpretation:





      1. $bigwedge^k_{mathbb{R}}mathbb{C}^d$: Here we take exterior power over $mathbb R$. In particular, we think of $mathbb{C}^d$ as a real $2d$-dimensional vector space. In that case $mathbb{R}^d$ is a vector subspace (over $mathbb{R}$), and so we have the general claim that “being decomposable” is a property which remains invariant under passing to a subspace. So, in that case, the answer is positive.


      Sasha's answer refers to this interpretation of the question.





      1. $bigwedge^k_{mathbb{C}}mathbb{C}^d$: Here we take exterior power over $mathbb C$, and we think of $mathbb{C}^d$ as a complex vector space. In this case $mathbb{R}^d$ is not a (complex) vector subspace of $mathbb{C}^d$, but we can still view $bigwedge^kmathbb{R}^d$ as a subspace of $bigwedge^k_{mathbb{C}}mathbb{C}^d$, via complexification. This because complexification commutes with exterior powers, so $(bigwedge^kmathbb{R}^d)^{mathbb C}=bigwedge^k_{mathbb{C}}((mathbb{R}^d)^{mathbb C})=bigwedge^k_{mathbb{C}}mathbb{C}^d.$ Any real vector space $V$ can be viewed as a subspace of its complexification $V^{mathbb C}=V otimes_{mathbb R}C$ via the map $v to v otimes 1$. Thus, we can consider in this way $bigwedge^kmathbb{R}^d$ as a subspace of $(bigwedge^kmathbb{R}^d)^{mathbb C}=bigwedge^k_{mathbb{C}}mathbb{C}^d.$
        Now we are given an element $omega in bigwedge^kmathbb{R}^d$, which is decomposable as an element in $bigwedge^k_{mathbb{C}}mathbb{C}^d$, and we ask whether or not it is decomposable as an element in $ bigwedge^kmathbb{R}^d$.



      What is the answer for this second variant of the question?




      Comment:



      Perhaps there is a way to view $bigwedge^k_{mathbb{C}}mathbb{C}^d$ as a subspace of $bigwedge^k_{mathbb{R}}mathbb{C}^d$ in a way which preserves decomposability, and the "real copy" $bigwedge^k_{mathbb{R}}mathbb{R}^d$, thus reducing the second problem to the first one. I asked about the possible existence of such an embedding here.



      So far, I only know that the answer is positive for $k=2$:



      Let $omega in bigwedge^2mathbb{R}^d$ be decomposable in $bigwedge^2mathbb{C}^d$. $omega$ can be written as



      $$omega=(u_1+iv_1) wedge (u_2+iv_2), tag{1}$$ where $u_1,u_2,v_1,v_2 in mathbb R^d$. Since



      $$ omega=(u_1 wedge u_2 - v_1 wedge v_2)+i (v_1 wedge u_2+u_1 wedge v_2), $$



      $omega in bigwedge^2mathbb{R}^d$ if and only if $$v_1 wedge u_2=-u_1 wedge v_2, tag{2}$$



      where this is an equality in $bigwedge^2mathbb{R}^d$.



      Suppose that $omega in bigwedge^2mathbb{R}^d$. If $dim(text{span}_{mathbb R}(u_1,u_2) cap text{span}_{mathbb R}(v_1,v_2)) ge 1$ then $omega=u_1 wedge u_2 - v_1 wedge v_2$ is decomposable in $bigwedge^2mathbb{R}^d$. Otherwise, $u_1,u_2,v_1,v_2$ are linearly independent over $mathbb R$, which violates equation $(2)$.



      I don't see an immediate generalization of this proof for $k ge 3$. By expanding $$omega=(u_1+iv_1) wedge (u_2+iv_2) wedge dots wedge (u_k+iv_k)$$
      we get more than two summands in the real part.










      share|cite|improve this question











      $endgroup$




      Let $k,d$ be positive integers, and let $omega in bigwedge^kmathbb{R}^d$ be decomposable in $ bigwedge^kmathbb{C}^d$.



      Is $omega$ decomposable in $bigwedge^kmathbb{R}^d$?



      Edit:



      Let me be more careful about the formulation of this question, as $bigwedge^kmathbb{C}^d$ can have two different non-isomorphic interpretation:





      1. $bigwedge^k_{mathbb{R}}mathbb{C}^d$: Here we take exterior power over $mathbb R$. In particular, we think of $mathbb{C}^d$ as a real $2d$-dimensional vector space. In that case $mathbb{R}^d$ is a vector subspace (over $mathbb{R}$), and so we have the general claim that “being decomposable” is a property which remains invariant under passing to a subspace. So, in that case, the answer is positive.


      Sasha's answer refers to this interpretation of the question.





      1. $bigwedge^k_{mathbb{C}}mathbb{C}^d$: Here we take exterior power over $mathbb C$, and we think of $mathbb{C}^d$ as a complex vector space. In this case $mathbb{R}^d$ is not a (complex) vector subspace of $mathbb{C}^d$, but we can still view $bigwedge^kmathbb{R}^d$ as a subspace of $bigwedge^k_{mathbb{C}}mathbb{C}^d$, via complexification. This because complexification commutes with exterior powers, so $(bigwedge^kmathbb{R}^d)^{mathbb C}=bigwedge^k_{mathbb{C}}((mathbb{R}^d)^{mathbb C})=bigwedge^k_{mathbb{C}}mathbb{C}^d.$ Any real vector space $V$ can be viewed as a subspace of its complexification $V^{mathbb C}=V otimes_{mathbb R}C$ via the map $v to v otimes 1$. Thus, we can consider in this way $bigwedge^kmathbb{R}^d$ as a subspace of $(bigwedge^kmathbb{R}^d)^{mathbb C}=bigwedge^k_{mathbb{C}}mathbb{C}^d.$
        Now we are given an element $omega in bigwedge^kmathbb{R}^d$, which is decomposable as an element in $bigwedge^k_{mathbb{C}}mathbb{C}^d$, and we ask whether or not it is decomposable as an element in $ bigwedge^kmathbb{R}^d$.



      What is the answer for this second variant of the question?




      Comment:



      Perhaps there is a way to view $bigwedge^k_{mathbb{C}}mathbb{C}^d$ as a subspace of $bigwedge^k_{mathbb{R}}mathbb{C}^d$ in a way which preserves decomposability, and the "real copy" $bigwedge^k_{mathbb{R}}mathbb{R}^d$, thus reducing the second problem to the first one. I asked about the possible existence of such an embedding here.



      So far, I only know that the answer is positive for $k=2$:



      Let $omega in bigwedge^2mathbb{R}^d$ be decomposable in $bigwedge^2mathbb{C}^d$. $omega$ can be written as



      $$omega=(u_1+iv_1) wedge (u_2+iv_2), tag{1}$$ where $u_1,u_2,v_1,v_2 in mathbb R^d$. Since



      $$ omega=(u_1 wedge u_2 - v_1 wedge v_2)+i (v_1 wedge u_2+u_1 wedge v_2), $$



      $omega in bigwedge^2mathbb{R}^d$ if and only if $$v_1 wedge u_2=-u_1 wedge v_2, tag{2}$$



      where this is an equality in $bigwedge^2mathbb{R}^d$.



      Suppose that $omega in bigwedge^2mathbb{R}^d$. If $dim(text{span}_{mathbb R}(u_1,u_2) cap text{span}_{mathbb R}(v_1,v_2)) ge 1$ then $omega=u_1 wedge u_2 - v_1 wedge v_2$ is decomposable in $bigwedge^2mathbb{R}^d$. Otherwise, $u_1,u_2,v_1,v_2$ are linearly independent over $mathbb R$, which violates equation $(2)$.



      I don't see an immediate generalization of this proof for $k ge 3$. By expanding $$omega=(u_1+iv_1) wedge (u_2+iv_2) wedge dots wedge (u_k+iv_k)$$
      we get more than two summands in the real part.







      differential-geometry algebraic-geometry complex-geometry exterior-algebra real-algebraic-geometry






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Feb 4 at 13:07







      Asaf Shachar

















      asked Feb 3 at 8:44









      Asaf ShacharAsaf Shachar

      5,79931145




      5,79931145






















          1 Answer
          1






          active

          oldest

          votes


















          6












          $begingroup$

          The question can be reformulated as follows: is
          $$
          Gr_{mathbb{C}}(k,d) cap mathbb{P}_{mathbb{R}}(wedge^k mathbb{R}^d) = Gr_{mathbb{R}}(k,d)?
          $$

          The answer is positive, because Plücker equations of the Grassmannian have real (in fact even integer) coefficients.



          Alternatively, this can be shown as follows. Decomposibility of $omega$ is equivalent to the fact that the dimension of the space of vectors annihilating $omega$ (by wedge products) is equal to $k$. This is equivalent to a rank condition for a system of linear equations with real coefficients, and its enough to note that its rank over $mathbb{R}$ is the same as its rank over $mathbb{C}$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you. I think that I was confused about some particular aspects of this problem, which are related to the the fact that we should distinguish whether we are taking the exterior (or tensor) power over $mathbb R$ or over $mathbb C$. I will try to clarify my confusion soon... (Anyway I think your answer is definitely relevant to whatever formulation I had in mind...).
            $endgroup$
            – Asaf Shachar
            Feb 3 at 11:56












          • $begingroup$
            (Note that TeX codes for diacritical marks don't work in math.SE)
            $endgroup$
            – Pedro Tamaroff
            Feb 4 at 13:06










          • $begingroup$
            Indeed, I have now clarified things. The question can be viewed via two different interpretations, depending on which field we are tensoring over. The less trivial case, which interests me more is when we tensor over $mathbb C$. I am quite sure that your answer is only applicable to the case where we tensor over $mathbb R$. I have updated the question thoroughly to include a discussion of the different cases, if you are interested. Thanks again for your help.
            $endgroup$
            – Asaf Shachar
            Feb 4 at 13:10












          Your Answer








          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "69"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3098324%2fif-omega-in-bigwedgek-mathbbrd-is-decomposable-over-mathbb-c-is-it%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          6












          $begingroup$

          The question can be reformulated as follows: is
          $$
          Gr_{mathbb{C}}(k,d) cap mathbb{P}_{mathbb{R}}(wedge^k mathbb{R}^d) = Gr_{mathbb{R}}(k,d)?
          $$

          The answer is positive, because Plücker equations of the Grassmannian have real (in fact even integer) coefficients.



          Alternatively, this can be shown as follows. Decomposibility of $omega$ is equivalent to the fact that the dimension of the space of vectors annihilating $omega$ (by wedge products) is equal to $k$. This is equivalent to a rank condition for a system of linear equations with real coefficients, and its enough to note that its rank over $mathbb{R}$ is the same as its rank over $mathbb{C}$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you. I think that I was confused about some particular aspects of this problem, which are related to the the fact that we should distinguish whether we are taking the exterior (or tensor) power over $mathbb R$ or over $mathbb C$. I will try to clarify my confusion soon... (Anyway I think your answer is definitely relevant to whatever formulation I had in mind...).
            $endgroup$
            – Asaf Shachar
            Feb 3 at 11:56












          • $begingroup$
            (Note that TeX codes for diacritical marks don't work in math.SE)
            $endgroup$
            – Pedro Tamaroff
            Feb 4 at 13:06










          • $begingroup$
            Indeed, I have now clarified things. The question can be viewed via two different interpretations, depending on which field we are tensoring over. The less trivial case, which interests me more is when we tensor over $mathbb C$. I am quite sure that your answer is only applicable to the case where we tensor over $mathbb R$. I have updated the question thoroughly to include a discussion of the different cases, if you are interested. Thanks again for your help.
            $endgroup$
            – Asaf Shachar
            Feb 4 at 13:10
















          6












          $begingroup$

          The question can be reformulated as follows: is
          $$
          Gr_{mathbb{C}}(k,d) cap mathbb{P}_{mathbb{R}}(wedge^k mathbb{R}^d) = Gr_{mathbb{R}}(k,d)?
          $$

          The answer is positive, because Plücker equations of the Grassmannian have real (in fact even integer) coefficients.



          Alternatively, this can be shown as follows. Decomposibility of $omega$ is equivalent to the fact that the dimension of the space of vectors annihilating $omega$ (by wedge products) is equal to $k$. This is equivalent to a rank condition for a system of linear equations with real coefficients, and its enough to note that its rank over $mathbb{R}$ is the same as its rank over $mathbb{C}$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you. I think that I was confused about some particular aspects of this problem, which are related to the the fact that we should distinguish whether we are taking the exterior (or tensor) power over $mathbb R$ or over $mathbb C$. I will try to clarify my confusion soon... (Anyway I think your answer is definitely relevant to whatever formulation I had in mind...).
            $endgroup$
            – Asaf Shachar
            Feb 3 at 11:56












          • $begingroup$
            (Note that TeX codes for diacritical marks don't work in math.SE)
            $endgroup$
            – Pedro Tamaroff
            Feb 4 at 13:06










          • $begingroup$
            Indeed, I have now clarified things. The question can be viewed via two different interpretations, depending on which field we are tensoring over. The less trivial case, which interests me more is when we tensor over $mathbb C$. I am quite sure that your answer is only applicable to the case where we tensor over $mathbb R$. I have updated the question thoroughly to include a discussion of the different cases, if you are interested. Thanks again for your help.
            $endgroup$
            – Asaf Shachar
            Feb 4 at 13:10














          6












          6








          6





          $begingroup$

          The question can be reformulated as follows: is
          $$
          Gr_{mathbb{C}}(k,d) cap mathbb{P}_{mathbb{R}}(wedge^k mathbb{R}^d) = Gr_{mathbb{R}}(k,d)?
          $$

          The answer is positive, because Plücker equations of the Grassmannian have real (in fact even integer) coefficients.



          Alternatively, this can be shown as follows. Decomposibility of $omega$ is equivalent to the fact that the dimension of the space of vectors annihilating $omega$ (by wedge products) is equal to $k$. This is equivalent to a rank condition for a system of linear equations with real coefficients, and its enough to note that its rank over $mathbb{R}$ is the same as its rank over $mathbb{C}$.






          share|cite|improve this answer











          $endgroup$



          The question can be reformulated as follows: is
          $$
          Gr_{mathbb{C}}(k,d) cap mathbb{P}_{mathbb{R}}(wedge^k mathbb{R}^d) = Gr_{mathbb{R}}(k,d)?
          $$

          The answer is positive, because Plücker equations of the Grassmannian have real (in fact even integer) coefficients.



          Alternatively, this can be shown as follows. Decomposibility of $omega$ is equivalent to the fact that the dimension of the space of vectors annihilating $omega$ (by wedge products) is equal to $k$. This is equivalent to a rank condition for a system of linear equations with real coefficients, and its enough to note that its rank over $mathbb{R}$ is the same as its rank over $mathbb{C}$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Feb 4 at 13:05









          Pedro Tamaroff

          97.7k10153299




          97.7k10153299










          answered Feb 3 at 9:24









          SashaSasha

          5,218139




          5,218139












          • $begingroup$
            Thank you. I think that I was confused about some particular aspects of this problem, which are related to the the fact that we should distinguish whether we are taking the exterior (or tensor) power over $mathbb R$ or over $mathbb C$. I will try to clarify my confusion soon... (Anyway I think your answer is definitely relevant to whatever formulation I had in mind...).
            $endgroup$
            – Asaf Shachar
            Feb 3 at 11:56












          • $begingroup$
            (Note that TeX codes for diacritical marks don't work in math.SE)
            $endgroup$
            – Pedro Tamaroff
            Feb 4 at 13:06










          • $begingroup$
            Indeed, I have now clarified things. The question can be viewed via two different interpretations, depending on which field we are tensoring over. The less trivial case, which interests me more is when we tensor over $mathbb C$. I am quite sure that your answer is only applicable to the case where we tensor over $mathbb R$. I have updated the question thoroughly to include a discussion of the different cases, if you are interested. Thanks again for your help.
            $endgroup$
            – Asaf Shachar
            Feb 4 at 13:10


















          • $begingroup$
            Thank you. I think that I was confused about some particular aspects of this problem, which are related to the the fact that we should distinguish whether we are taking the exterior (or tensor) power over $mathbb R$ or over $mathbb C$. I will try to clarify my confusion soon... (Anyway I think your answer is definitely relevant to whatever formulation I had in mind...).
            $endgroup$
            – Asaf Shachar
            Feb 3 at 11:56












          • $begingroup$
            (Note that TeX codes for diacritical marks don't work in math.SE)
            $endgroup$
            – Pedro Tamaroff
            Feb 4 at 13:06










          • $begingroup$
            Indeed, I have now clarified things. The question can be viewed via two different interpretations, depending on which field we are tensoring over. The less trivial case, which interests me more is when we tensor over $mathbb C$. I am quite sure that your answer is only applicable to the case where we tensor over $mathbb R$. I have updated the question thoroughly to include a discussion of the different cases, if you are interested. Thanks again for your help.
            $endgroup$
            – Asaf Shachar
            Feb 4 at 13:10
















          $begingroup$
          Thank you. I think that I was confused about some particular aspects of this problem, which are related to the the fact that we should distinguish whether we are taking the exterior (or tensor) power over $mathbb R$ or over $mathbb C$. I will try to clarify my confusion soon... (Anyway I think your answer is definitely relevant to whatever formulation I had in mind...).
          $endgroup$
          – Asaf Shachar
          Feb 3 at 11:56






          $begingroup$
          Thank you. I think that I was confused about some particular aspects of this problem, which are related to the the fact that we should distinguish whether we are taking the exterior (or tensor) power over $mathbb R$ or over $mathbb C$. I will try to clarify my confusion soon... (Anyway I think your answer is definitely relevant to whatever formulation I had in mind...).
          $endgroup$
          – Asaf Shachar
          Feb 3 at 11:56














          $begingroup$
          (Note that TeX codes for diacritical marks don't work in math.SE)
          $endgroup$
          – Pedro Tamaroff
          Feb 4 at 13:06




          $begingroup$
          (Note that TeX codes for diacritical marks don't work in math.SE)
          $endgroup$
          – Pedro Tamaroff
          Feb 4 at 13:06












          $begingroup$
          Indeed, I have now clarified things. The question can be viewed via two different interpretations, depending on which field we are tensoring over. The less trivial case, which interests me more is when we tensor over $mathbb C$. I am quite sure that your answer is only applicable to the case where we tensor over $mathbb R$. I have updated the question thoroughly to include a discussion of the different cases, if you are interested. Thanks again for your help.
          $endgroup$
          – Asaf Shachar
          Feb 4 at 13:10




          $begingroup$
          Indeed, I have now clarified things. The question can be viewed via two different interpretations, depending on which field we are tensoring over. The less trivial case, which interests me more is when we tensor over $mathbb C$. I am quite sure that your answer is only applicable to the case where we tensor over $mathbb R$. I have updated the question thoroughly to include a discussion of the different cases, if you are interested. Thanks again for your help.
          $endgroup$
          – Asaf Shachar
          Feb 4 at 13:10


















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3098324%2fif-omega-in-bigwedgek-mathbbrd-is-decomposable-over-mathbb-c-is-it%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          android studio warns about leanback feature tag usage required on manifest while using Unity exported app?

          'app-layout' is not a known element: how to share Component with different Modules

          SQL update select statement