Abstract
We give an explicit description of the Mirković-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV polytopes. We prove that a polytope is an MV polytope if and only if it is a lattice polytope whose defining hyperplanes are parallel to those of the Weyl polytopes and whose 2-faces are rank 2 MV polytopes. As an application, we give a bijection between Lusztig’s canonical basis and the set of MV polytopes.
-
[jared1] J. E. Anderson, "On Mirković and Vilonen’s Intersection Homology cycles for the Loop Grassmannian," PhD Thesis , Princeton University, 2000.
@phdthesis{jared1,
author={Anderson, Jared E.},
TITLE={On Mirković and Vilonen's Intersection Homology cycles for the Loop Grassmannian},
SCHOOL={Princeton University},
YEAR={2000},
} -
[jared2]
J. E. Anderson, "A polytope calculus for semisimple groups," Duke Math. J., vol. 116, iss. 3, pp. 567-588, 2003.
@article {jared2, MRKEY = {1958098},
AUTHOR = {Anderson, Jared E.},
TITLE = {A polytope calculus for semisimple groups},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {116},
YEAR = {2003},
NUMBER = {3},
PAGES = {567--588},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {20G05 (14L99)},
MRNUMBER = {2004a:20047},
MRREVIEWER = {Timo Neuvonen},
DOI = {10.1215/S0012-7094-03-11636-1},
ZBLNUMBER = {1064.20047},
} -
[jaredmisha]
J. Anderson and M. Kogan, "Mirković-Vilonen cycles and polytopes in Type A," Internat. Math. Res. Not., iss. 12, pp. 561-591, 2004.
@article {jaredmisha, MRKEY = {2040622},
AUTHOR = {Anderson, Jared and Kogan, Mikhail},
TITLE = {Mirković-{V}ilonen cycles and polytopes in {T}ype {A}},
JOURNAL = {Internat. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
YEAR = {2004},
NUMBER = {12},
PAGES = {561--591},
ISSN = {1073-7928},
MRCLASS = {14C25 (14M25)},
MRNUMBER = {2005d:14013},
MRREVIEWER = {Baohua Fu},
DOI = {10.1155/S1073792804132765},
ZBLNUMBER = {1082.14012},
} -
[jaredmisha2] J. Anderson and M. Kogan, "The algebra of Mirković-Vilonen cycles in type A," Pure Appl. Math. Q., vol. 2, iss. 4, part 2, pp. 1187-1215, 2006.
@article {jaredmisha2, MRKEY = {2282418},
AUTHOR = {Anderson, Jared and Kogan, Mikhail},
TITLE = {The algebra of {M}irković-{V}ilonen cycles in type {A}},
JOURNAL = {Pure Appl. Math. Q.},
FJOURNAL = {Pure and Applied Mathematics Quarterly},
VOLUME = {2},
YEAR = {2006},
NUMBER = {4, part 2},
PAGES = {1187--1215},
ISSN = {1558-8599},
MRCLASS = {14L35 (14C25)},
MRNUMBER = {2007j:14071},
MRREVIEWER = {Dmitrii A. Timash{ë}v},
ZBLNUMBER = {1110.14043},
} -
[BD] A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves.
@misc{BD,
author={Beilinson, A. and Drinfeld, V.},
TITLE={Quantization of Hitchin's integrable system and Hecke eigensheaves},
NOTE={{\tt http://www.math.uchicago.edu/\verb'~'arinkin/langlands/}},
} -
[BFZA]
A. Berenstein, S. Fomin, and A. Zelevinsky, "Parametrizations of canonical bases and totally positive matrices," Adv. Math., vol. 122, iss. 1, pp. 49-149, 1996.
@article {BFZA, MRKEY = {1405449},
AUTHOR = {Berenstein, Arkady and Fomin, Sergey and Zelevinsky, Andrei},
TITLE = {Parametrizations of canonical bases and totally positive matrices},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {122},
YEAR = {1996},
NUMBER = {1},
PAGES = {49--149},
ISSN = {0001-8708},
CODEN = {ADMTA4},
MRCLASS = {17B37 (05E15 20G05)},
MRNUMBER = {98j:17008},
MRREVIEWER = {Kailash C. Misra},
DOI = {10.1006/aima.1996.0057},
ZBLNUMBER = {0966.17011},
} -
[BFZ2]
A. Berenstein, S. Fomin, and A. Zelevinsky, "Cluster algebras. III. Upper bounds and double Bruhat cells," Duke Math. J., vol. 126, iss. 1, pp. 1-52, 2005.
@article {BFZ2, MRKEY = {2110627},
AUTHOR = {Berenstein, Arkady and Fomin, Sergey and Zelevinsky, Andrei},
TITLE = {Cluster algebras. {III}. {U}pper bounds and double {B}ruhat cells},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {126},
YEAR = {2005},
NUMBER = {1},
PAGES = {1--52},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {16S99 (05E15 14M17 22E46)},
MRNUMBER = {2005i:16065},
DOI = {10.1215/S0012-7094-04-12611-9},
ZBLNUMBER = {1135.16013},
} -
[BZschub]
A. Berenstein and A. Zelevinsky, "Total positivity in Schubert varieties," Comment. Math. Helv., vol. 72, iss. 1, pp. 128-166, 1997.
@article {BZschub, MRKEY = {1456321},
AUTHOR = {Berenstein, Arkady and Zelevinsky, Andrei},
TITLE = {Total positivity in {S}chubert varieties},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {72},
YEAR = {1997},
NUMBER = {1},
PAGES = {128--166},
ISSN = {0010-2571},
CODEN = {COMHAX},
MRCLASS = {14M15 (22E46)},
MRNUMBER = {99g:14064},
MRREVIEWER = {Kailash C. Misra},
DOI = {10.1007/PL00000363},
ZBLNUMBER = {0891.20030},
} -
[BZtpm]
A. Berenstein and A. Zelevinsky, "Tensor product multiplicities, canonical and totally positive varieties," Invent. Math., vol. 143, iss. 1, pp. 77-128, 2001.
@article {BZtpm, MRKEY = {1802793},
AUTHOR = {Berenstein, Arkady and Zelevinsky, Andrei},
TITLE = {Tensor product multiplicities, canonical and totally positive varieties},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {143},
YEAR = {2001},
NUMBER = {1},
PAGES = {77--128},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {17B10 (12K10 17B20 17B37 20G05)},
MRNUMBER = {2002c:17005},
MRREVIEWER = {Ivan V. Arzhantsev},
ZBLNUMBER = {1061.17006},
DOI = {10.1007/s002220000102},
} -
[BG]
A. Braverman and D. Gaitsgory, "Crystals via the affine Grassmannian," Duke Math. J., vol. 107, iss. 3, pp. 561-575, 2001.
@article {BG, MRKEY = {1828302},
AUTHOR = {Braverman, Alexander and Gaitsgory, Dennis},
TITLE = {Crystals via the affine {G}rassmannian},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {107},
YEAR = {2001},
NUMBER = {3},
PAGES = {561--575},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {20G05 (14F05)},
MRNUMBER = {2002e:20083},
MRREVIEWER = {Andrew Mathas},
DOI = {10.1215/S0012-7094-01-10736-9},
ZBLNUMBER = {1015.20030},
} -
[BFG] A. Braverman, M. Finkelberg, and D. Gaitsgory, "Uhlenbeck spaces via affine Lie algebras," in The Unity of Mathematics, Boston, MA: Birkhäuser, 2006, pp. 17-135.
@incollection {BFG, MRKEY = {2181803},
AUTHOR = {Braverman, Alexander and Finkelberg, Michael and Gaitsgory, Dennis},
TITLE = {Uhlenbeck spaces via affine {L}ie algebras},
BOOKTITLE = {The Unity of Mathematics},
SERIES = {Progr. Math.},
NUMBER = {244},
PAGES = {17--135},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {2006},
MRCLASS = {14D20 (17B67)},
MRNUMBER = {2007f:14008},
MRREVIEWER = {Ivan S. Kausz},
ZBLNUMBER = {1105.14013},
} -
[E] G. Ewald, Combinatorial Convexity and Algebraic Geometry, New York: Springer-Verlag, 1996.
@book {E, MRKEY = {1418400},
AUTHOR = {Ewald, G{ü}nter},
TITLE = {Combinatorial Convexity and Algebraic Geometry},
SERIES = {Grad. Texts Math.},
NUMBER = {168},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1996},
PAGES = {xiv+372},
ISBN = {0-387-94755-8},
MRCLASS = {52B20 (14M25 52B05)},
MRNUMBER = {97i:52012},
MRREVIEWER = {P. McMullen},
ZBLNUMBER = {0869.52001},
} -
[FM] M. Finkelberg and I. Mirković, "Semi-infinite flags. I. Case of global curve $\Bbb P^1$," in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Providence, RI: Amer. Math. Soc., 1999, pp. 81-112.
@incollection {FM, MRKEY = {1729360},
AUTHOR = {Finkelberg, Michael and Mirkovi{ć},
Ivan},
TITLE = {Semi-infinite flags. {I}. {C}ase of global curve {$\bold P\sp 1$}},
BOOKTITLE = {Differential Topology, Infinite-Dimensional {L}ie Algebras, and Applications},
SERIES = {Amer. Math. Soc. Transl. Ser. 2},
NUMBER = {194},
PAGES = {81--112},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1999},
MRCLASS = {14F43 (14M15 20G10 20G42)},
MRNUMBER = {2001j:14029},
MRREVIEWER = {Alexander E. Polishchuk},
ZBLNUMBER = {1076.14512},
} -
[BF] E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Second ed., Providence, RI: Amer. Math. Soc., 2004.
@book {BF, MRKEY = {2082709},
AUTHOR = {Frenkel, Edward and Ben-Zvi, David},
TITLE = {Vertex Algebras and Algebraic Curves},
SERIES = {Math. Surveys Monogr.},
NUMBER = {88},
EDITION = {Second},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2004},
PAGES = {xiv+400},
ISBN = {0-8218-3674-9},
MRCLASS = {17B69 (14D21 14H81)},
MRNUMBER = {2005d:17035},
ZBLNUMBER = {1106.17035},
} -
[FZ]
S. Fomin and A. Zelevinsky, "Double Bruhat cells and total positivity," J. Amer. Math. Soc., vol. 12, iss. 2, pp. 335-380, 1999.
@article {FZ, MRKEY = {1652878},
AUTHOR = {Fomin, Sergey and Zelevinsky, Andrei},
TITLE = {Double {B}ruhat cells and total positivity},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {12},
YEAR = {1999},
NUMBER = {2},
PAGES = {335--380},
ISSN = {0894-0347},
MRCLASS = {20G20 (15A23)},
MRNUMBER = {2001f:20097},
DOI = {10.1090/S0894-0347-99-00295-7},
ZBLNUMBER = {0913.22011},
} -
[F] W. Fulton, Introduction to Toric Varieties, Princeton, NJ: Princeton Univ. Press, 1993.
@book {F, MRKEY = {1234037},
AUTHOR = {Fulton, William},
TITLE = {Introduction to Toric Varieties},
SERIES = {Ann. of Math. Studies},
NUMBER = {131},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1993},
PAGES = {xii+157},
ISBN = {0-691-00049-2},
MRCLASS = {14M25 (14-02 14J30)},
MRNUMBER = {94g:14028},
MRREVIEWER = {T. Oda},
ZBLNUMBER = {0813.14039},
} -
[GL]
S. Gaussent and P. Littelmann, "LS galleries, the path model, and MV cycles," Duke Math. J., vol. 127, iss. 1, pp. 35-88, 2005.
@article {GL, MRKEY = {2126496},
AUTHOR = {Gaussent, S. and Littelmann, P.},
TITLE = {L{S} galleries, the path model, and {MV} cycles},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {127},
YEAR = {2005},
NUMBER = {1},
PAGES = {35--88},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {20G05 (14M15 17B10 22E46 51E24)},
MRNUMBER = {2006c:20092},
MRREVIEWER = {Guy Rousseau},
DOI = {10.1215/S0012-7094-04-12712-5},
ZBLNUMBER = {1078.22007},
} -
[G] V. Ginzburg, Perverse sheaves on a loop group and Langlands duality.
@misc{G,
author={Ginzburg, V.},
TITLE={Perverse sheaves on a loop group and Langlands duality},
NOTE={available at math.AG/9511007},
} -
[MVcrystal]
J. Kamnitzer, "The crystal structure on the set of Mirković-Vilonen polytopes," Adv. Math., vol. 215, iss. 1, pp. 66-93, 2007.
@article {MVcrystal, MRKEY = {2354986},
AUTHOR = {Kamnitzer, Joel},
TITLE = {The crystal structure on the set of {M}irković-{V}ilonen polytopes},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {215},
YEAR = {2007},
NUMBER = {1},
PAGES = {66--93},
ISSN = {0001-8708},
CODEN = {ADMTA4},
MRCLASS = {17B37},
MRNUMBER = {2009a:17021},
MRREVIEWER = {Jae-Hoon Kwon},
DOI = {10.1016/j.aim.2007.03.012},
ZBLNUMBER = {1134.14028},
} -
[L] G. Lusztig, "Singularities, character formulas, and a $q$-analog of weight multiplicities," in Analysis and Topology on Singular Spaces, II, III (Luminy, 1981), Paris: Soc. Math. France, 1983, pp. 208-229.
@incollection {L, MRKEY = {737932},
AUTHOR = {Lusztig, George},
TITLE = {Singularities, character formulas, and a {$q$}-analog of weight multiplicities},
BOOKTITLE = {Analysis and Topology on Singular Spaces, {\rm II, III} ({L}uminy, 1981)},
SERIES = {Astérisque},
NUMBER = {101},
PAGES = {208--229},
PUBLISHER = {Soc. Math. France},
ADDRESS = {Paris},
YEAR = {1983},
MRCLASS = {17B10 (05A30 20G05 22E47)},
MRNUMBER = {85m:17005},
MRREVIEWER = {James E. Humphreys},
ZBLNUMBER = {0561.22013},
} -
[Lbook]
G. Lusztig, "Canonical bases arising from quantized enveloping algebras," J. Amer. Math. Soc., vol. 3, iss. 2, pp. 447-498, 1990.
@article {Lbook, MRKEY = {1035415},
AUTHOR = {Lusztig, George},
TITLE = {Canonical bases arising from quantized enveloping algebras},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {3},
YEAR = {1990},
NUMBER = {2},
PAGES = {447--498},
ISSN = {0894-0347},
MRCLASS = {17B35 (16A64)},
MRNUMBER = {90m:17023},
MRREVIEWER = {Ya. S. So{\u\i}bel{\cprime}man},
DOI = {10.2307/1990961},
ZBLNUMBER = {0703.17008},
} -
[Lproj] G. Lusztig, "Introduction to quantized enveloping algebras," in New Developments in Lie Theory and their Applications (Córdoba, 1989), Boston, MA: Birkhäuser, 1992, pp. 49-65.
@incollection {Lproj, MRKEY = {1190735},
AUTHOR = {Lusztig, George},
TITLE = {Introduction to quantized enveloping algebras},
BOOKTITLE = {New Developments in {L}ie Theory and their Applications ({C}órdoba, 1989)},
SERIES = {Progr. Math.},
NUMBER = {105},
PAGES = {49--65},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1992},
MRCLASS = {17B37 (17B10 20F36 81R50)},
MRNUMBER = {94b:17029},
MRREVIEWER = {Seok-Jin Kang},
ZBLNUMBER = {0767.17014},
} -
[Lbookbook] G. Lusztig, Introduction to Quantum Groups, Boston, MA: Birkhäuser, 1993.
@book {Lbookbook, MRKEY = {1227098},
AUTHOR = {Lusztig, George},
TITLE = {Introduction to Quantum Groups},
SERIES = {Progr. Math.},
NUMBER = {110},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1993},
PAGES = {xii+341},
ISBN = {0-8176-3712-5},
MRCLASS = {17B37 (16W30 17-02 17B35 81R50)},
MRNUMBER = {94m:17016},
MRREVIEWER = {Jie Du},
ZBLNUMBER = {0788.17010},
} -
[Lfunnypaper]
G. Lusztig, "An algebraic-geometric parametrization of the canonical basis," Adv. Math., vol. 120, iss. 1, pp. 173-190, 1996.
@article {Lfunnypaper, MRKEY = {1392278},
AUTHOR = {Lusztig, George},
TITLE = {An algebraic-geometric parametrization of the canonical basis},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {120},
YEAR = {1996},
NUMBER = {1},
PAGES = {173--190},
ISSN = {0001-8708},
CODEN = {ADMTA4},
MRCLASS = {17B37 (17B10 17B20)},
MRNUMBER = {97g:17015},
MRREVIEWER = {Kailash C. Misra},
DOI = {10.1006/aima.1996.0036},
ZBLNUMBER = {0877.17006},
} -
[MV] I. Mirković and K. Vilonen, "Perverse sheaves on affine Grassmannians and Langlands duality," Math. Res. Lett., vol. 7, iss. 1, pp. 13-24, 2000.
@article {MV, MRKEY = {1748284},
AUTHOR = {Mirkovi{ć},
Ivan and Vilonen, Kari},
TITLE = {Perverse sheaves on affine {G}rassmannians and {L}anglands duality},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {7},
YEAR = {2000},
NUMBER = {1},
PAGES = {13--24},
ISSN = {1073-2780},
MRCLASS = {14F43 (20G05 32S60)},
MRNUMBER = {2001h:14020},
MRREVIEWER = {Alexander E. Polishchuk},
ZBLNUMBER = {0987.14015},
} -
[MV2]
I. Mirković and K. Vilonen, "Geometric Langlands duality and representations of algebraic groups over commutative rings," Ann. of Math., vol. 166, iss. 1, pp. 95-143, 2007.
@article {MV2, MRKEY = {2342692},
AUTHOR = {Mirkovi{ć},
Ivan and Vilonen, Kari},
TITLE = {Geometric {L}anglands duality and representations of algebraic groups over commutative rings},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {166},
YEAR = {2007},
NUMBER = {1},
PAGES = {95--143},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {22E55 (11R39 20G05)},
MRNUMBER = {2008m:22027},
MRREVIEWER = {Peter Fiebig},
DOI = {10.4007/annals.2007.166.95},
ZBLNUMBER = {1138.22013},
} -
[speyer]
D. E. Speyer, "Horn’s problem, Vinnikov curves, and the hive cone," Duke Math. J., vol. 127, iss. 3, pp. 395-427, 2005.
@article {speyer, MRKEY = {2132865},
AUTHOR = {Speyer, David E.},
TITLE = {Horn's problem, {V}innikov curves, and the hive cone},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {127},
YEAR = {2005},
NUMBER = {3},
PAGES = {395--427},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {14P99 (05E15 15A42)},
MRNUMBER = {2005m:14113},
MRREVIEWER = {Thomas C. Craven},
DOI = {10.1215/S0012-7094-04-12731-0},
ZBLNUMBER = {1069.14037},
} -
[SS]
D. Speyer and B. Sturmfels, "The tropical Grassmannian," Adv. Geom., vol. 4, iss. 3, pp. 389-411, 2004.
@article {SS, MRKEY = {2071813},
AUTHOR = {Speyer, David and Sturmfels, Bernd},
TITLE = {The tropical {G}rassmannian},
JOURNAL = {Adv. Geom.},
FJOURNAL = {Advances in Geometry},
VOLUME = {4},
YEAR = {2004},
NUMBER = {3},
PAGES = {389--411},
ISSN = {1615-715X},
CODEN = {AGDEA3},
MRCLASS = {14P99 (13J30 14M25)},
MRNUMBER = {2005d:14089},
MRREVIEWER = {Meirav Amram-Blei},
DOI = {10.1515/advg.2004.023},
ZBLNUMBER = {1065.14071},
} -
[Z] G. M. Ziegler, Lectures on Polytopes, New York: Springer-Verlag, 1995.
@book {Z, MRKEY = {1311028},
AUTHOR = {Ziegler, G{ü}nter M.},
TITLE = {Lectures on Polytopes},
SERIES = {Grad. Texts Math.},
NUMBER = {152},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1995},
PAGES = {x+370},
ISBN = {0-387-94365-X},
MRCLASS = {52Bxx},
MRNUMBER = {96a:52011},
MRREVIEWER = {Margaret M. Bayer},
ZBLNUMBER = {0823.52002},
}