ALGEBRAIC COMBINATORICS

no. 4
Tropical positivity and determinantal varieties
Brandenburg, Marie-Charlotte1; Loho, Georg2; Sinn, Rainer3
1 Max Planck Institute for Mathematics in the Sciences Inselstraße 22 04103 Leipzig (Germany)
2 University of Twente Drienerlolaan 5 7522 NB Enschede (Netherlands)
3 Universität Leipzig Augustusplatz 10 04109 Leipzig (Germany)
Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 999-1040.

We initiate the study of positive-tropical generators as positive analogues of the concept of tropical bases. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. We focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. Moreover, in the case of square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.286
Classification: 14T15, 14P05, 14M12, 05E14, 52B40
Keywords: signed tropicalization, positive-tropical generators, determinantal varieties, Birkhoff polytope, bicolored phylogenetic trees
Brandenburg, Marie-Charlotte 1; Loho, Georg 2; Sinn, Rainer 3

1 Max Planck Institute for Mathematics in the Sciences Inselstraße 22 04103 Leipzig (Germany)
2 University of Twente Drienerlolaan 5 7522 NB Enschede (Netherlands)
3 Universität Leipzig Augustusplatz 10 04109 Leipzig (Germany)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Brandenburg, Marie-Charlotte; Loho, Georg; Sinn, Rainer. Tropical positivity and determinantal varieties. Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 999-1040. doi : 10.5802/alco.286. https://alco.centre-mersenne.org/articles/10.5802/alco.286/

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