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Total Positivity

どの になるような totally positive ばれる

Lusztig [ Lus94 ] 典的 total positivity real reductive group totally positive part した Lusztig による total positivity [ Lus98 Lus08 ] ある

  • tatally nonnetative Grassmannian
  • totally nonnegative flag variety

それらの cell については , Fomin Zelevinsky [ FZ99 ] Rietsch [ Rie99 ] など により 研究 されている

Grassmann には , Postnikov [ Pos ] しらべて いる

Williams [ Wil07 ] totally nonnegative flag variety regular CW complex であ ることを している Postnikov [ Pos ] Introduction Grassmannian closed ball regular CW complex つことを して いる

Rietsch Williams [ RW10 ] discrete Morse theory いて flag variety cell closure collapsible であることを している

Grassmannian cell CW であることは , Postnikov, Speyer, Williams [ PSW09 ] している Galashin, Karp, Lam [ GKL ] , totally nonnegative Grassmannian closed ball であることを した , して いる

B. Keller [ Kel11 ] Fomin [ Fom10 ] によると , Fomin Zelevinsky cluster algebra えた つは , total positivity する アプロ けることだ たようである

Fomin [ Fom10 ] によると total positivity のようなことと があるようであ :

  • classical mechanics,
  • probability,
  • discrete potential theory,
  • asymptotic representation theory,
  • algebraic and enumerative combinatorics,
  • linear algebra and its applications.

Kodama Williams [ KW11 KW14 ] によると , KP equation とも ある

Arkani-Hamed Trnka [ AHT14b AHT14a ] 導入 した amplintuhedron , Grassmannian totally nonnegative part ( から された Grassmannian ) による として されるが , その motivation super Yang-Mills theory scattering amplitude 研究 にあるところが Lam [ Lam16 ] により tree amplituhedron cohomology class 調 べられて いる

References

[AHT14a]     Nima Arkani-Hamed and Jaroslav Trnka. Into the Amplituhedron. JHEP , 12:182, 2014, arXiv:1312.7878 .

[AHT14b]     Nima Arkani-Hamed and Jaroslav Trnka. The Amplituhedron. JHEP , 10:030, 2014, arXiv:1312.2007 .

[Fom10]     Sergey Fomin. Total positivity and cluster algebras. In Proceedings of the International Congress of Mathematicians. Volume II , pages 125–145, New Delhi, 2010. Hindustan Book Agency, arXiv:1005.1086 .

[FZ99]     Sergey Fomin and Andrei Zelevinsky. Double Bruhat cells and total positivity. J. Amer. Math. Soc. , 12(2):335–380, 1999, http://dx.doi.org/10.1090/S0894-0347-99-00295-7 .

[GKL]     Pavel Galashin, Steven N. Karp, and Thomas Lam. The totally nonnegative Grassmannian is a ball, arXiv:1707.02010 .

[Kel11]     Bernhard Keller. Categorification of acyclic cluster algebras: an introduction. In Higher structures in geometry and physics , volume 287 of Progr. Math. , pages 227–241. Birkhäuser/Springer, New York, 2011, arXiv:0801.3103 .

[KW11]     Yuji Kodama and Lauren K. Williams. KP solitons, total positivity, and cluster algebras. Proc. Natl. Acad. Sci. USA , 108(22):8984–8989, 2011, arXiv:1105.4170 .

[KW14]     Yuji Kodama and Lauren Williams. KP solitons and total positivity for the Grassmannian. Invent. Math. , 198(3):637–699, 2014, arXiv:1106.0023 .

[Lam16]     Thomas Lam. Amplituhedron cells and Stanley symmetric functions. Comm. Math. Phys. , 343(3):1025–1037, 2016, arXiv:1408.5531 .

[Lus94]     G. Lusztig. Total positivity in reductive groups. In Lie theory and geometry , volume 123 of Progr. Math. , pages 531–568. Birkhäuser Boston, Boston, MA, 1994, https://doi.org/10.1007/978-1-4612-0261-5_20 .

[Lus98]     George Lusztig. Introduction to total positivity. In Positivity in Lie theory: open problems , volume 26 of de Gruyter Exp. Math. , pages 133–145. de Gruyter, Berlin, 1998.

[Lus08]     G. Lusztig. A survey of total positivity. Milan J. Math. , 76:125–134, 2008, arXiv:0705.3842 .

[Pos]     Alexander Postnikov. Total positivity, Grassmannians, and networks, arXiv:math/0609764 .

[PSW09]     Alexander Postnikov, David Speyer, and Lauren Williams. Matching polytopes, toric geometry, and the totally non-negative Grassmannian. J. Algebraic Combin. , 30(2):173–191, 2009, arXiv:0706.2501 .

[Rie99]     Konstanze Rietsch. An algebraic cell decomposition of the nonnegative part of a flag variety. J. Algebra , 213(1):144–154, 1999, arXiv:alg-geom/9709035 .

[RW10]     Konstanze Rietsch and Lauren Williams. Discrete Morse theory for totally non-negative flag varieties. Adv. Math. , 223(6):1855–1884, 2010, arXiv:0810.4314 .

[Wil07]     Lauren K. Williams. Shelling totally nonnegative flag varieties. J. Reine Angew. Math. , 609:1–21, 2007, arXiv:math/0509129 .