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AN ALGORITHM TO CONSTRUCT THE LE DIAGRAM ASSOCIATED TO A GRASSMANN NECKLACE

Part of: Algebraic combinatorics Designs and configurations

Published online by Cambridge University Press:  18 January 2019

SUSAMA AGARWALA  and
SIÂN FRYER
SUSAMA AGARWALA
Affiliation:
Johns Hopkins Applied Physics Lab, Laurel, MD 20723, USA e-mail: [email protected]
SIÂN FRYER*
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106-3080, USA e-mail: [email protected]
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Abstract

Le diagrams and Grassmann necklaces both index the collection of positroids in the nonnegative Grassmannian Gr≥0(k, n), but they excel at very different tasks: for example, the dimension of a positroid is easily extracted from its Le diagram, while the list of bases of a positroid is far more easily obtained from its Grassmann necklace. Explicit bijections between the two are, therefore, desirable. An algorithm for turning a Le diagram into a Grassmann necklace already exists; in this note, we give the reverse algorithm.

MSC classification

Primary: 05B35: Matroids, geometric lattices
Secondary: 05E10: Combinatorial aspects of representation theory
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 1 , January 2020 , pp. 85 - 91
Creative Commons
This is a work of the U.S. Government and is not subject to copyright protection in the United States
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019

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References

Agarwala, S. and Amat, E. M., Wilson loop diagrams and positroids, Commun. Math. Phys. 350(2) (2016), 569–601.Google Scholar
Agarwala, S. and Fryer, S., A study in ${\mathbb{G}_{\mathbb{R},+}}$(2, 6): from the geometric case book of Wilson loop diagrams and SYM N = 4, arXiv:1803.00958.Google Scholar
Arkani-Hamed, N. and Trnka, N. J., The amplituhedron, J. High Energ. Phys. 2014(10) (2014).Google Scholar
Casteels, K. and Fryer, S., From Grassmann necklaces to decorated permutations and back again, Algebr. Represent. Theory 20(4) (2017), 895–921.CrossRefGoogle Scholar
Casteels, K., Quantum matrices by paths, Algebr. Number Theory 8(8) (2014), 1857–1912.CrossRefGoogle Scholar
Cauchon, G., Spectre premier de O q(M n(k)): image canonique et séparation normale, J. Algebra 260(2) (2003), 519–569.CrossRefGoogle Scholar
Goodearl, K. R., Launois, S. and Lenagan, T. H., Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves, Math. Z. 269(1–2) (2011), 29–45.CrossRefGoogle Scholar
Karp, S. N., Williams, L. K. and Zhang, Y. X., Decompositions of amplituhedra, arXiv:1708.09525.Google Scholar
Kodama, Y. and Williams, L., KP solitons and total positivity for the Grassmannian, Invent. Math. 198(3) (2014), 637–699.CrossRefGoogle Scholar
Launois, S., Lenagan, T. H. and Rigal, L., Prime ideals in the quantum Grassmannian, Selecta Math. (N.S.) 13(4) (2008), 697–725.CrossRefGoogle Scholar
Oh, S., Positroids and Schubert matroids, J. Combin. Theory Ser. A 118(8) (2011), 2426–2435.CrossRefGoogle Scholar
Postnikov, A., Total positivity, Grassmannians, and networks, arXiv:math/0609764.Google Scholar