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Totally nonnegative cells and matrix Poisson varieties

Totally nonnegative cells and matrix Poisson varieties,10.1016/j.aim.2010.07.010,Advances in Mathematics,K. R. Goodearl,S. Launois,T. H. Lenagan

Totally nonnegative cells and matrix Poisson varieties   (Citations: 7)
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We describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices, that is, the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by A. Postnikov. In order to do this, we relate the totally nonnegative cells to torus orbits of symplectic leaves of the Poisson varieties of complex matrices. In particular, we describe the minors that vanish on a torus orbit of symplectic leaves, we prove that such families of minors are exactly the admissible families, and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices.
Journal: Advances in Mathematics - ADVAN MATH , vol. 226, no. 1, pp. 779-826, 2011
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    • ...Goodearl, Launois and Lenagan [11] independently find all minors that belong to the vanishing ideal of the Zariski closure of any T -orbit of symplectic leaves S(y) in (Mm,n, πm,n)...

    Milen Yakimov. Invariant prime ideals in quantizations of nilpotent Lie algebras

    • ...In view of the close connections that have been discovered between totally nonnegative matrices, the standard Poisson matrix variety and quantum matrices, see, for example, [2, 3], and between the totally nonnegative grassmannian and the quantum grassmannian,...

    S Launoiset al. Twisting the quantum grassmannian

    • ...Connections between the second and third of these objects were developed in [9]...
    • ...In [9], we study this connection, and prove that a family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative matrices if and only if this family is exactly the set of minors that vanish on the closure of a certain torus-orbit of symplectic leaves in the matrix Poisson variety...
    • ...Mm,p(C) have been explicitly described in [9], based on results of Fulton [7] and Brown-Goodearl-Yakimov [2]...
    • ...In [9, Theorem 2.9], we extended the results of the previous theorem...
    • ...In [9, Section 5], we constructed an (explicit) bijection between the set of m◊p Cauchon diagrams and the set of Poisson H-primes in O(Mm,p(C))...
    • ...In [9, Section 5], we constructed an (explicit) bijection between the set of m◊p Cauchon diagrams and the set of Poisson H-primes in O(Mm,p(C)). As in [9, Theorem 5.3], we denote by J0...
    • ...Also, in the case where K = C, it was proved in [9, Theorem 5.4] that, if J0 is a Poisson...
    • ...However, it is given in [9, Section 5] using the (Poisson) restoration algorithm (the inverse of the deleting-derivations algorithm), in the following way...
    • ...C. Note that the induction step mimics the proof of [9, Theorem 3.16]...
    • ...Proof. The equivalence of (1) and (2) is proved in [9, Theorem 6.2]...
    • ...The families of minors that vanish on the closure of an H-orbit of symplectic leaves have been explicitly described in [9, Theorem 2.11]...
    • ...Proof. (1) is a consequence of Theorem 4.2 and [9, Theorem 2.11]...

    K. R. Goodearlet al. Torus-invariant prime ideals in quantum matrices, totally nonnegative ...

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