Totally nonnegative cells and matrix Poisson varieties

Goodearl, K. R.; Launois, Stéphane; Lenagan, T. H.

doi:10.1016/j.aim.2010.07.010

Cited by 32 publications

(60 citation statements)

References 20 publications

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“…Connections between the second and third of these objects were developed in [9]. Here we close the circle by linking the first and second objects.…”

Section: Introductionmentioning

confidence: 99%

Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

2010

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The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size. As a corollary, we obtain explicit generating sets of quantum minors for the torus-invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over Q.

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“…Connections between the second and third of these objects were developed in [9]. Here we close the circle by linking the first and second objects.…”

Section: Introductionmentioning

confidence: 99%

Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

2010

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“…An alternative to the existing proofs [4, p. 62] of this fact relies on the restoration algorithm [14] which is the inverse of the Cauchon algorithm. Let A be T N .…”

Section: I) a Is T P (T N ) (Ii) Complete Neville Elimination Appliementioning

confidence: 99%

“…Cauchon diagrams and the Cauchon algorithm. In this subsection, we first recall from [14], [16] the definition of a Cauchon diagram and of the Cauchon algorithm 1 .…”

Section: Elamentioning

confidence: 99%

“…In contrast to [14], [16], we do not start from a fixed Cauchon diagram but sequentially construct the lacunary sequences. It is sufficient to describe the construction of the sequence from the starting pair (i 0 , j 0 ) to the next pair (i 1 , j 1 ) with 0 <ã i1,j1 since for a given matrix A the determinantal test is performed by moving row by row from the bottom to the top row.…”

Section: ) With Respect To C Starting At This Entry Assume That Eitmentioning

confidence: 99%

“…For properties of these matrices the reader is referred to the two recently published monographs [4], [17]. This paper grew out our study of the papers [14], [16]. Herein the authors apply the so-called Cauchon diagrams to parameterize the set of the totally nonnegative matrices with a fixed pattern of zero minors (if this set is nonempty), called a totally nonnegative cell (corresponding to this pattern).…”

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Improved tests and characterizations of totally nonnegative matrices

Adm¹,

Garloff²

2014

ELA

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Abstract. Totally nonnegative matrices, i.e., matrices having all minors nonnegative, are considered. A condensed form of the Cauchon algorithm which has been proposed for finding a parameterization of the set of these matrices with a fixed pattern of vanishing minors is derived. The close connection of this variant to Neville elimination and bidiagonalization is shown and new determinantal tests for total nonnegativity are developed which require much fewer minors to be checked than for the tests known so far. New characterizations of some subclasses of the totally nonnegative matrices as well as shorter proofs for some classes of matrices for being (nonsingular and) totally nonnegative are derived.

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From Totally Nonnegative Matrices to Quantum Matrices and Back, via Poisson Geometry

Launois

Lenagan

2017

Perspectives in Lie Theory

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In this survey article, we describe recent work that connects three separate objects of interest: totally nonnegative matrices; quantum matrices; and matrix Poisson varieties. on submatrices. The classification is given in terms of certain permutations from the relevant symmetric group with restrictions arising from the Bruhat order.The nonnegative part of the space of real matrices consists of those matrices whose minors are all nonnegative. One can specify a cell decomposition of the set of totally nonnegative matrices by specifying exactly which minors are to be zero/nonzero. In [34], Postnikov classified the nonempty cells by means of a bijection with certain diagrams, known as Le-diagrams. The work of Postnikov was then developed by Talaska [36], Williams [37], etc., and led to the definition of positroid varieties that have been recently studied by Oh [32] and Knutson, Lam and Speyer [23].The interesting observation from the point of view of this work is that in each of the above three sets of results the combinatorial objects that arise turn out to be the same! The definitions of Cauchon diagrams and Le-diagrams are the same, and the restricted permutations arising in the Brown-Goodearl-Yakimov study can be seen to lead to Cauchon/Le diagrams via the notion of pipe dreams.Once one is aware of these connections, this suggests that there should be a connection between torus-invariant prime ideals, torus-orbits of symplectic leaves and totally nonnegative cells. This connection has been investigated in recent papers by Goodearl and the present authors, [14,15]. In particular, we have shown that the Restoration Algorithm, developed by the first author for use in quantum matrices, can also be used in the other two settings to answer questions concerning the torusorbits of symplectic leaves and totally nonnegative cells. The detailed proofs of the results that were obtained in [14,15] are very technical, and our aim in this survey, is to describe the results informally and to compute some examples to illuminate our results.

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Totally nonnegative cells and matrix Poisson varieties

Cited by 32 publications

References 20 publications

Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

Improved tests and characterizations of totally nonnegative matrices

From Totally Nonnegative Matrices to Quantum Matrices and Back, via Poisson Geometry

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