Rank 2 affine MV polytopes

Baumann, Pierre; Dunlap, Thomas; Kamnitzer, Joel; Tingley, Peter

doi:10.1090/S1088-4165-2013-00438-7

Rank 2 affine MV polytopes
HTML articles powered by AMS MathViewer

by Pierre Baumann, Thomas Dunlap, Joel Kamnitzer and Peter Tingley PDF

Represent. Theory 17 (2013), 442-468 Request permission

Abstract:

We give a realization of the crystal $B(-\infty )$ for $\widehat {\mathrm {sl}}_2$ using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito’s characterization of $B(-\infty )$, in terms of the $*$ involution. The polygons we use have combinatorial properties suggesting they are the $\widehat {\mathrm {sl}}_2$ analogues of the Mirković-Vilonen polytopes defined by Anderson and the third author in finite type. Using Kashiwara’s similarity of crystals we also give MV polytopes for $A_2^{(2)}$, the other rank 2 affine Kac-Moody algebra.

References

Jared E. Anderson, A polytope calculus for semisimple groups, Duke Math. J. 116 (2003), no. 3, 567–588. MR 1958098, DOI 10.1215/S0012-7094-03-11636-1
Pierre Baumann, Joel Kamnitzer, and Peter Tingley. Affine Mirković-Vilonen polytopes. Preprint. arXiv:1110.3661.
Jonathan Beck, Vyjayanthi Chari, and Andrew Pressley, An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (1999), no. 3, 455–487. MR 1712630, DOI 10.1215/S0012-7094-99-09915-5
Ilaria Damiani, A basis of type Poincaré-Birkhoff-Witt for the quantum algebra of $\widehat \textrm {sl}(2)$, J. Algebra 161 (1993), no. 2, 291–310. MR 1247357, DOI 10.1006/jabr.1993.1220
Thomas Rough Dunlap II, Combinatorial representation theory of affine sl2 via polytope calculus, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)–Northwestern University. MR 2736801
Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002. MR 1881971, DOI 10.1090/gsm/042
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Joel Kamnitzer, The crystal structure on the set of Mirković-Vilonen polytopes, Adv. Math. 215 (2007), no. 1, 66–93. MR 2354986, DOI 10.1016/j.aim.2007.03.012
Joel Kamnitzer, Mirković-Vilonen cycles and polytopes, Ann. of Math. (2) 171 (2010), no. 1, 245–294. MR 2630039, DOI 10.4007/annals.2010.171.245
Masaki Kashiwara, The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839–858. MR 1240605, DOI 10.1215/S0012-7094-93-07131-1
Masaki Kashiwara, On crystal bases, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 155–197. MR 1357199
Masaki Kashiwara, Similarity of crystal bases, Lie algebras and their representations (Seoul, 1995) Contemp. Math., vol. 194, Amer. Math. Soc., Providence, RI, 1996, pp. 177–186. MR 1395599, DOI 10.1090/conm/194/02393
Masaki Kashiwara and Yoshihisa Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997), no. 1, 9–36. MR 1458969, DOI 10.1215/S0012-7094-97-08902-X
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
Dinakar Muthiah, Double MV cycles and the Naito-Sagaki-Saito crystal, Adv. Math. 240 (2013), 268–290. MR 3046309, DOI 10.1016/j.aim.2013.02.019
Dinakar Muthiah and Peter Tingley, Affine PBW bases and MV polytopes in rank $2$, Selecta Mathematica, published on line January 24, 2013, DOI 10.1007/s0029-012-0117-z.
Satoshi Naito, Daisuke Sagaki, and Yoshihisa Saito, Toward Berenstein-Zelevinsky data in affine type $A$, part I: Construction of the affine analogs, Algebraic groups and quantum groups, Contemp. Math., vol. 565, Amer. Math. Soc., Providence, RI, 2012, pp. 143–184. MR 2932426, DOI 10.1090/conm/565/11180