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We describe explicitly the admissible families of minors for the totallynonnegative cells of real matrices, that is, the families of minors thatproduce nonempty cells in the cell decompositions of spaces of totallynonnegative matrices introduced by A. Postnikov. In order to do this, we relatethe totally nonnegative cells to torus orbits of symplectic leaves of thePoisson varieties of complex matrices. In particular, we describe the minorsthat vanish on a torus orbit of symplectic leaves, we prove that such familiesof minors are exactly the admissible families, and we show that the nonemptytotally nonnegative cells are the intersections of the torus orbits ofsymplectic leaves with the spaces of totally nonnegative matrices.

Totally nonnegative cells and matrix Poisson varieties