Degenerate conformal blocks for the W₃ algebra at c=2 and Specht polynomials

We study a homogeneous system of d+8 linear partial differential equations (PDEs) in d variables arising from two-dimensional Conformal Field Theories (CFTs) with a W₃-symmetry algebra. In the CFT context, d PDEs are third order and correspond to the null-state equations, whereas the remaining 8 PDEs (five being second order and three being first order) correspond to the W₃ global Ward identities. In the case of central charge c=2, we construct a subspace of the space of all solutions which grow no faster than a power law. We call this subspace the space of W₃ conformal blocks, and we provide a basis expressed in terms of Specht polynomials associated with column-strict, rectangular Young tableaux with three columns. The dimension of this space is a Kotska number and it coincides with CFT predictions, hence we conjecture that it exhausts the space of all solutions having a power law bound. Moreover, we prove that the space of W₃ conformal blocks is an irreducible representation of a certain diagram algebra defined from sl₃ webs that we call Kuperberg algebra. Finally, we formulate a precise conjecture relating the W₃ conformal blocks at c=2 to scaling limits of probabilities in the triple dimer model recently studied by Kenyon and Shi. We verify the conjecture for explicit examples up to d=6. For more general central charges, we expect that W₃ conformal blocks are related to scaling limits of probabilities in lattice models based on sl₃ webs.