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In the paper, the super Grassmannian $G_{2|2}(4|4)$ is defined by (12)--(18).

An element of $G_{2|2}(4|4)$ can be written as a $(2 | 2) \times (4 | 4)$ matrix of full rank modulo the left action by an invertible $(2 | 2) \times (2 | 2)$ matrix. The definition of full rank for supermatrices is there is an invertible $(2 | 2) \times (2 | 2)$ submatrix.

My questions are:

  1. Can we write an element in $G_{2|2}(4|4)$ as follows? \begin{align} \left( \begin{array}{c c c c | c c c c} x_{11} & x_{12} & x_{13} & x_{14} & x_{15} & x_{16} & x_{17} & x_{18} \\ x_{21} & x_{22} & x_{23} & x_{24} & x_{25} & x_{26} & x_{27} & x_{28} \\ \hline x_{31} & x_{32} & x_{33} & x_{34} & x_{35} & x_{36} & x_{37} & x_{38} \\ x_{41} & x_{42} & x_{43} & x_{44} & x_{45} & x_{46} & x_{47} & x_{48} \end{array} \right) \end{align}

  2. What are the differences between the Grassmannian $G(4,8)$ and super Grassmannian $G_{2|2}(4|4)$? I think that one difference is that the entries of matrices in $G_{2|2}(4|4)$ are even or odd and the entries of matrices in $G(4,8)$ are all even.

Thank you very much.

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  • $\begingroup$ Comment to the post (v4): It seems you basically answered your own questions. $\endgroup$
    – Qmechanic ♦
    Commented Jun 10, 2016 at 11:06

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