## Abstract

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 \({\mathbb{Q}}\).

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## Acknowledgements

The results in this paper were announced during the mini-workshop “Non-negativity is a quantum phenomenon” that took place at the Mathematisches Forschungsinstitut Oberwolfach, 1–7 March 2009, [20]; we thank the director and staff of the MFO for providing the ideal environment for this stimulating meeting. We also thank Konni Rietsch, Laurent Rigal, Lauren Williams and Milen Yakimov for discussions and comments concerning this paper both at the workshop and at other times. Finally, we thank Karel Casteels for sending us his preprint [3].

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The research of K. R. Goodearl was supported by a grant from the National Science Foundation (USA).

The research of S. Launois was supported by a Marie Curie European Reintegration Grant within the 7th European Community Framework Programme.

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**Open Access** This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Goodearl, K.R., Launois, S. & Lenagan, T.H. Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves.
*Math. Z.* **269**, 29–45 (2011). https://doi.org/10.1007/s00209-010-0714-5

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DOI: https://doi.org/10.1007/s00209-010-0714-5

### Keywords

- Quantum matrices
- Torus-invariant prime ideals
- Quantum minors
- Totally nonnegative cells
- Symplectic leaves