tropical Plucker relations

The following are copied from Kamnitzer’s paper (see Sources), and he indicates that they come from Berenstein-Zelevinsky.

We say that a collection (M_\gamma)_{\gamma\in\Gamma} satisfies the tropical Plucker relation at (w,i,j) where w\in W;i,j\in I such that ws_i>w, ws_j>w, i\neq j if a_{ij}=0 or if

  1. if a_{ji}=a_{ij}=-1 then
    M_{ws_i\cdot\Lambda_i} + M_{ws_j\cdot\Lambda_j} = \min(M_{w\cdot\Lambda_i} + M_{ws_is_j\cdot\Lambda_j}, M_{ws_js_i\cdot\Lambda_i} + M_{w\cdot\Lambda_j});
  2. if a_{ij}=-1,a_{ji}=-2 then
    M_{ws_j\cdot\Lambda_j} + M_{ws_is_j\cdot\Lambda_j} + M_{ws_i\cdot\Lambda_i}=\min ( 2M_{ws_is_j\cdot\Lambda_j} + M_{w\cdot\Lambda_i},
      2M_{w\cdot\Lambda_j} + M_{ws_is_js_i\cdot\Lambda_i},
      M_{w\cdot\Lambda_j} + M_{ws_js_is_j\cdot\Lambda_j} + M_{ws_i\cdot\Lambda_i}),
    M_{ws_js_i\cdot\Lambda_i} + 2M_{ws_is_j\cdot\Lambda_j} + M_{ws_i\cdot\Lambda_i}=\min ( 2M_{w\cdot\Lambda_j} + 2M_{ws_is_js_i\cdot\Lambda_i},
      2M_{ws_js_is_j\cdot\Lambda_j} + 2M_{ws_i\cdot\Lambda_i},
      M_{ws_is_js_i\cdot\Lambda_i} + 2M_{ws_is_j\cdot\Lambda_j} + M_{w\cdot\Lambda_i}),
  3. if a_{ij}=-2, a_{ji}=-1, then
    4. M_{ws_js_i\cdot\Lambda_i} + M_{ws_i\cdot\Lambda_i} + M_{ws_is_j\cdot\Lambda_j}=\min ( 2M_{ws_i\cdot\Lambda_i} + M_{ws_js_is_j\cdot\Lambda_j},
      2M_{ws_is_js_i\cdot\Lambda_i} + M_{w\cdot\Lambda_j},
      M_{ws_is_js_i\cdot\Lambda_i} + M_{w\cdot\Lambda_i} + M_{ws_is_j\cdot\Lambda_j}),
    M_{ws_j\cdot\Lambda_j} + 2M_{ws_i\cdot\Lambda_i} + M_{ws_is_j\cdot\Lambda_j}=\min ( 2M_{ws_is_js_i\cdot\Lambda_i} + 2M_{w\cdot\Lambda_j},
      2M_{w\cdot\Lambda_i} + 2M_{ws_is_j\cdot\Lambda_j},
      M_{w\cdot\Lambda_j} + 2M_{ws_i\cdot\Lambda_i} + M_{ws_js_is_j\cdot\Lambda_j}),

And we simply say the collection satisfies the tropical Plucker relations if it satisfies them at all (w,i,j) described above.

4 Responses to “tropical Plucker relations”

  1. DN Says:
    December 9, 2007 at 8:11 am | Reply

    I’d really like to understand how these equations are derived. Namely, starting from the ordinary Plucker relations, how does one tropicalize?

  2. trdunlap2 Says:
    December 12, 2007 at 8:37 am | Reply

    Its seems that tropicalizing a function is very easy but tropicalizing ideals is somewhat subtle.

  3. trdunlap2 Says:
    December 12, 2007 at 6:05 pm | Reply

    As for tropicalization, here’s a for example. Take x_1\lambda_1-x_2\lambda_2+x_3\lambda_3=0. If we tropicalize naively by taking valuations then we might get something like:
    \min\{\tilde{x}_1+\tilde\lambda_1,\tilde{x}_2+\tilde\lambda_2,\tilde{x}_3+\tilde\lambda_3\}\le\infty
    But that’s no relation at all. So what we do, and I don’t understand the process 100% yet, is consider a different form of the equation. For example:
    x_1\lambda_1=x_2\lambda_2-x_3\lambda_3
    yields:
    \tilde x_1+\tilde\lambda_1=\min\{\tilde x_2+\tilde\lambda_2,\tilde x_3+\tilde\lambda_3\}
    This is what I get so far from the Speyer/Sturmfels paper I added to my “sources” page.

  4. trdunlap2 Says:
    December 12, 2007 at 7:16 pm | Reply

    I should say, though, that the equations in Kamnitzer are pulled directly from a paper by Berenstein and Zelevinsky (arXiv:math.RT/9912012) in the “naive” way. So I my impression from S&S may be wrong.

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