A periodicity theorem for the octahedron recurrence
André Henriques
Westfälische Wilhelms-Universität Mathematisches Institut Einsteinstr. 62 48149 Münster Germany
DOI: 10.1007/s10801-006-0045-0
Abstract
The octahedron recurrence lives on a 3-dimensional lattice and is given by f( x, y, t+1)=( f( x+1, y, t) f( x -1, y, t)+ f( x, y+1, t) f( x, y -1, t))/ f( x, y, t -1) f(x,y,t+1)=(f(x+1,y,t)f(x-1,y,t)+f(x,y+1,t)f(x,y-1,t))/f(x,y,t-1). In this paper, we investigate a variant of this recurrence which lives in a lattice contained in [0, m] \times [0, n] \times \mathbb R [0,m] \times [0,n] \times \mathbb R. Following Speyer, we give an explicit non-recursive formula for the values of this recurrence and use it to prove that it is periodic of period n+ m. We then proceed to show various other hidden symmetries satisfied by this bounded octahedron recurrence.
Pages: 1–26
Keywords: keywords octahedron recurrence; Laurent phenomenon; perfect matchings
Full Text: PDF
References
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2. S. Fomin and A. Zelevinsky, “The Laurent phenomenon,” Adv. in Appl. Math. 28(2) (2002), 119-144.
3. S. Fomin and A. Zelevinsky, “Y -systems and generalized associahedra,” Ann. of Math. (2), 158(3) (2003), 977-1018.
4. A. Henriques and J. Kamnitzer, “The octahedron recurrence and gl(n) crystals,” to appear in Adv. in Math., available at Math.QA/0408114, 2005.
5. W.H. Mills, D.P. Robbins, and H. Rumsey Jr, “Alternating sign matrices and descending plane partitions,” J. Combin. Theory Ser. A 34(3) (1983), 340-359.
6. J. Propp, “The many faces of alternating-sign matrices,” Discrete Math. Theor. Comput. Sci. Proc. AA (2001), 43-58.
7. J. Propp, Enumeration of Matchings: Problems and Progress, Volume 38 of New Perspectives in Algebraic Combinatorics. Cambridge Univ. Press, Cambridge, 1999, pp. 255-291.
8. D.P. Robbins and H. Rumsey Jr, “Determinants and alternating sign matrices,” Adv. in Math. 62(2) (1986), 169-184.
9. D.E. Speyer, “Perfect matchings and the octahedron recurrence,” to appear in J. Algebraic Combin. Math.CO/0402452, 2004.