ar X iv : m at h - ph / 0 61 20 18 v 1 6 D ec 2 00 6 BKP PLANE PARTITIONS

Using BKP neutral fermions, we derive a product expression for the generating function of volume-weighted plane partitions that satisfy two conditions. If we call a set of adjacent equal height-h columns, h > 0, an h-path, then 1. Every h-path can assume one of two possible colours. 2. There is a unique way to move along an h-path from any column to another. In [1], Okounkov and Reshetikhin observed that certain charged free fermion vertex operators can be used to generate plane partitions. In [2], with Vafa, they used this observation to compute the partition function of a topological string theory. As these vertex operators arise in KP theory [3], it is natural to look for analogous results in the context of other integrable hierarchies [4]. Definition 1. An 'h-path' in a plane partition, or simply a 'path' when indicating the height h is not needed, is a set of adjacent equal height-h columns, where h > 0. In this note, we use BKP neutral free fermion vertex operators to obtain the generating function of volume-weighted plane partitions [5], that satisfy two conditions. 1. Every h-path, h > 0, can assume one of two possible colours, so it contributes a factor of 2 to the multiplicity of the plane partition, irrespective of h. 2. There is a unique way to move along an h-path, from one column to another, or equivalently 'every h-path is 1-column wide'. Definition 2. A 'BKP plane partition' is a plane partition that satisfies the above two conditions. Example 1. In Figure 1, we use tableau-like notation to represent a plane partition of the type counted in this note. The integers are the column heights. The volume of a plane partition is the sum of all column heights. The volume in this case is 39. There are 6 h-paths. Each path can assume one of two possible colours, so the multiplicity of this plane partition is 2 6 = 64. 2 1 3 3 3 6 4 3 6 5 3 Figure 1. A tableau-like representation of a plane partition. There is a 3-path of length 5, a 6-path of length 2, and 4 different height paths of length 1 each. There is a unique way to move from any column on a path to another column. Counting a no-move on a length-1 h-path as the one (and only) possible move, every h-path is 1-column wide, …