Enumeration of non-positive planar trivalent graphs
Bruce W. Westbury
University of Warwick Mathematics Institute Coventry CV4 7AL UK
DOI: 10.1007/s10801-006-0041-4
Abstract
In this paper we construct inverse bijections between two sequences of finite sets. One sequence is defined by planar diagrams and the other by lattice walks. In [13] it is shown that the number of elements in these two sets are equal. This problem and the methods we use are motivated by the representation theory of the exceptional simple Lie algebra G 2. However in this account we have emphasised the combinatorics.
Pages: 357–373
Keywords: planar graphs; lattice paths; invariant tensors
Full Text: PDF
References
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2. S. Doty, “Presenting generalized q-Schur algebras,” Represent. Theory 7 (2003), 196-213 (electronic), arXiv:math.QA/0305208.
3. I.B. Frenkel and M.G. Khovanov, “Canonical bases in tensor products and graphical calculus for Uq (sl2),” Duke Math. J. 87(3) (1997), 409-480.
4. Z. Gao and N.C. Wormald, “Enumeration of rooted cubic planar maps,” Ann. Comb. 6(3-4) (2002), 313-325.
5. M. Kashiwara, “Crystalizing the q-analogue of universal enveloping algebras,” Comm. Math. Phys. 133(2) (1990), 249-260.
6. M. Kashiwara, “On crystal bases of the q-analogue of universal enveloping algebras,” Duke Math. J. 63(2) (1991), 465-516.
7. M. Khovanov and G. Kuperberg, “Web bases for sl(3) are not dual canonical,” Pacific J. Math. 188(1) (1999), 129-153, arXiv:q-alg/9712046.
8. S.-J. Kang, M. Kashiwara, K.C. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki, “Affine crystals and vertex models,” in Infinite Analysis, Part A, B (Kyoto, 1991), volume 16 of Adv. Ser. Math. Phys., World Sci. Publishing, River Edge, NJ, 1992, pp. 449-484.
9. S.-J. Kang and K.C. Misra, “Crystal bases and tensor product decompositions of Uq (G2)-modules,” J. Algebra 163(3) (1994), 675-691.
10. G. Kuperberg, “Spiders for rank 2 Lie algebras,” Comm. Math. Phys. 180(1) (1996), 109-151, arXiv:q- alg/9712003.
11. C. Lecouvey, “Schensted type correspondence for type G2 and computation of the canonical basis of a finite dimensional Uq (G2)-module”, arXiv:math.CO/0211443.
12. P. Littelmann, “A plactic algebra for semisimple Lie algebras,” Adv. Math. 124(2) (1996), 312-331.
13. K.R. Parthasarathy, R. Ranga Rao, and V.S. Varadarajan, “Representations of complex semi-simple Lie groups and Lie algebras,” Ann. of Math. (2) 85 (1967), 383-429.
14. N.J. Sloane, “The on-line encylopedia of integer sequences”, 2006. http://www.research.att.com/njas/ sequences/.
15. J.R. Stembridge, “Combinatorial models for Weyl characters,” Adv. Math. 168(1) (2002), 96-131.
16. W.T. Tutte, “A census of planar triangulations,” Canad. J. Math. 14 (1962), 21-38.
17. B.W. Westbury, “Invariant tensors for the spin representation of so(7)”, arXiv:math.QA/0601209.
18. S. Yamane, “Perfect crystals of Uq (G(1)),” J. Algebra 210(2) (1998), 440-486, arXiv:q-alg/9712012.