$f (n, i, j) f (n-2, i, j) = f (n-1, i-1, j) f (n-1, i + 1, j)
+ \lambda f (n-1, i, j-1) f (n-1, i, j + 1)$

This is a recurrence obtained in Robbins, Rumsey 1986 by extension of Desnanot-Jacobi (Lewis-Caroll) identity underlying the recursive computation of determinants called Dodgson condensation method which is obtained by setting $\lambda = 1$, $f(-1,i,j)=1$ and allowing only $n+i+j = 0\,\,\,(mod\,\,2)$.

Solutions are Laurent polynomials in $f(0,i,j)$ and $\lambda$. This is an example in cluster algebra theory.

According to Robbins, Rumsey 1986, exponents of the $f(0,i,j)$ in any monomial occurring in a $f(n_0,i_0,j_0)$ form pairs of compatible alternating sign matrices.

Birational Robinson–Schensted–Knuth correspondence satisfies the octahedron recurrence.

Appearance in the study of alternating-sign matrices

- D. P. Robbins, H. Rumsey,
*Determinants and alternating-sign matrices*, Advances in Math.**62**(1986) 169–184 doi - J. Propp,
*The many faces of alternating-sign matrices, in Discrete Models: Combinatorics, Computation, and Geometry*, Discrete Math. Theor. Comput. Sci. Proc., AA, Maison Inform. Math. Discrét. (Paris, 2001) 43–58

Providing a major example in cluster algebras

- Sergey Fomin, Andrei Zelevinsky,
*The Laurent phenomenon*, Adv. in Appl. Math. 28 (2002), no. 2, 119–144 - David E. Speyer,
*Perfect matchings and the octahedron recurrence*, J. Algebr. Comb. (2007) 25:309–348 doi arXiv:math.CO/0402452 - V. Danilov, Gleb Koshevoy,
*Arrays and the octahedron recurrence*, arXiv:math.CO/0504299 - André Henriques,
*A periodicity theorem for the octahedron recurrence*, J. Algebraic Comb. 26(1), 1–26 (2007) arXiv:math.CO/0604289

Relation to crystal bases

- André Henriques, Joel Kamnitzer,
*The octahedron recurrence and $gl_n$ crystals*, Adv. Math.**206**:1 (2006) 211–249 doi

Application to Littlewood-Richardson rule

- Allen Knutson, Terence Tao, Christopher T. Woodward.
*A positive proof of the Littlewood-Richardson rule using the octahedron recurrence*, arXiv:math.CO/0306274

Robinson–Schensted–Knuth correspondence satisfies octahedron recurrence

- V.I. Danilov, G. A. Koshevoy,
*The octahedron recurrence and RSK-correspondence*, Sém. Lothar. Combin., 54A (2005/07) Art. B54An, 16 pp. (electronic) published pdf arXiv:math.CO/0703414

On a bijective proof of that fact

- Miriam Farber, Sam Hopkins, Wuttisak Trongsiriwat,
*Interlacing networks: Birational RSK, the octahedron recurrence, and Schur function identities*, Journal of Combinatorial Theory**A 133**(2015) 339–371 doi

category: combinatorics

Last revised on August 2, 2024 at 19:23:55. See the history of this page for a list of all contributions to it.