Defining amplituhedra and Grassmann polytopes
The totally nonnegative Grassmannian Gr≥0 k,n is the set of k-dimensional subspaces V of Rn whose nonzero Plucker coordinates all have the same sign. In their study of scattering amplitudes in N = 4 supersym- metric Yang-Mills theory, Arkani-Hamed and Trnka (2013) considered the image (called an amp...
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Discrete Mathematics & Theoretical Computer Science
2020-04-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/6356/pdf |
| _version_ | 1797270253327089664 |
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| author | Steven N. Karp |
| author_facet | Steven N. Karp |
| author_sort | Steven N. Karp |
| collection | DOAJ |
| description | The totally nonnegative Grassmannian Gr≥0 k,n is the set of k-dimensional subspaces V of Rn whose nonzero Plucker coordinates all have the same sign. In their study of scattering amplitudes in N = 4 supersym- metric Yang-Mills theory, Arkani-Hamed and Trnka (2013) considered the image (called an amplituhedron) of Gr≥0 k,n under a linear map Z : Rn → Rr, where k ≤ r and the r × r minors of Z are all positive. One reason they required this positivity condition is to ensure that the map Gr≥0 k,n → Grk,r induced by Z is well defined, i.e. it takes everynelement of Gr≥0 k,n to a k-dimensional subspace of Rr. Lam (2015) gave a sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in which case he called the image a Grassmann polytope. (In the case k = 1, Grassmann polytopes are just polytopes, and amplituhedra are cyclic polytopes.) We give a necessary and sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in terms of sign variation. Using previous work we presented at FPSAC 2015, we obtain an equivalent condition in terms of the r × r minors of Z (assuming Z has rank r). |
| first_indexed | 2024-04-25T02:01:20Z |
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| id | doaj.art-fecacd3494cd4ac896e3e2ffcce5430a |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:01:20Z |
| publishDate | 2020-04-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-fecacd3494cd4ac896e3e2ffcce5430a2024-03-07T14:55:20ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502020-04-01DMTCS Proceedings, 28th...10.46298/dmtcs.63566356Defining amplituhedra and Grassmann polytopesSteven N. Karp0Department of Mathematics [Berkeley]The totally nonnegative Grassmannian Gr≥0 k,n is the set of k-dimensional subspaces V of Rn whose nonzero Plucker coordinates all have the same sign. In their study of scattering amplitudes in N = 4 supersym- metric Yang-Mills theory, Arkani-Hamed and Trnka (2013) considered the image (called an amplituhedron) of Gr≥0 k,n under a linear map Z : Rn → Rr, where k ≤ r and the r × r minors of Z are all positive. One reason they required this positivity condition is to ensure that the map Gr≥0 k,n → Grk,r induced by Z is well defined, i.e. it takes everynelement of Gr≥0 k,n to a k-dimensional subspace of Rr. Lam (2015) gave a sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in which case he called the image a Grassmann polytope. (In the case k = 1, Grassmann polytopes are just polytopes, and amplituhedra are cyclic polytopes.) We give a necessary and sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in terms of sign variation. Using previous work we presented at FPSAC 2015, we obtain an equivalent condition in terms of the r × r minors of Z (assuming Z has rank r).https://dmtcs.episciences.org/6356/pdf[math.math-co]mathematics [math]/combinatorics [math.co] |
| spellingShingle | Steven N. Karp Defining amplituhedra and Grassmann polytopes Discrete Mathematics & Theoretical Computer Science [math.math-co]mathematics [math]/combinatorics [math.co] |
| title | Defining amplituhedra and Grassmann polytopes |
| title_full | Defining amplituhedra and Grassmann polytopes |
| title_fullStr | Defining amplituhedra and Grassmann polytopes |
| title_full_unstemmed | Defining amplituhedra and Grassmann polytopes |
| title_short | Defining amplituhedra and Grassmann polytopes |
| title_sort | defining amplituhedra and grassmann polytopes |
| topic | [math.math-co]mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/6356/pdf |
| work_keys_str_mv | AT stevennkarp definingamplituhedraandgrassmannpolytopes |