Abstract
We introduce a recurrence which we term the multidimensional cube recurrence, generalizing the octahedron
recurrence studied by Propp, Fomin and Zelevinsky, Speyer, and Fock and Goncharov and the
three-dimensional cube recurrence studied by Fomin and Zelevinsky, and Carroll and Speyer. The states
of this recurrence are indexed by tilings of a polygon with rhombi, and the variables in the recurrence are
indexed by vertices of these tilings. We travel from one state of the recurrence to another by performing
elementary flips. We show that the values of the recurrence are independent of the order in which we perform
the flips; this proof involves nontrivial combinatorial results about rhombus tilings which may be of
independent interest. We then show that the multidimensional cube recurrence exhibits the Laurent phenomenon
– any variable is given by a Laurent polynomial in the other variables. We recognize a special
case of the multidimensional cube recurrence as giving explicit equations for the isotropic Grassmannians
IG(n − 1, 2n). Finally, we describe a tropical version of the multidimensional cube recurrence and show
that, like the tropical octahedron recurrence, it propagates certain linear inequalities.
Original language | English |
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Pages (from-to) | 1107-1136 |
Number of pages | 30 |
Journal | Advances in Mathematics |
Volume | 223 |
Issue number | 3 |
DOIs | |
Publication status | Published - 2010 |