nLab positroid

Definition

A positroid is a matroid represented by a $k\times n$ matrix whose maximal minors are nonnegative.

Properties

Positive Grassmannians have decomposition into positroid cells found by A. Postnikov. Thus the positroid cells play role in the calculation of scattering amplitudes in supersymmetric Yang–Mills (Arkani-Hamed et al. 2021).

Literature

Related $n$Lab entries include Grassmann necklace (a combinatorial object used for parametrizing positroid cells) and total positivity.

The notion is introduced in the seminal work

• Alexander Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764

On the connection between prime spectra of quantum Grassmannian?s and positroid cells in nonnegative Grassmannian:

• S. Launois, T. H. Lenagan, B. M. Nolan, Total positivity is a quantum phenomenon: the Grassmannian case, Memoirs of the Amer. Math. Soc. 1448 (2023) 123 p. Zbl07767960

Other works

• Federico Ardila, F. Rincon, L. Williams, Positroids and non-crossing partitions, Trans. Amer. Math. Soc. 368 (2016), no. 1, 337–363

• Suho Oh, Positroids and Schubert matroids, J. Combin. Theory A 118:8 (2011) 2426–2435 arXiv:0803.1018 doi

• Nima Arkani-Hamed, Thomas Lam, Marcus Spradlin, Positive configuration space, Commun. Math. Phys. 384, 909–954 (2021) doi

An analogue for flag matroids and flag varieties

• Jonathan Boretsky, The tropical geometry of flag positroids, PhD Thesis, Harvard 2024 pdf

category: combinatorics, algebra