Three Combinatorial Models for $$ _ $$ Crystals, with Applications to Cylindric Plane Partitions

Abstract

Abstract. We define three combinatorial models for b sln crystals, parametrized by partitions, configurations of beads on an "abacus", and cylindric plane partitions, respectively. These are reducible, but we can identify an irreducible subcrystal corresponding to any dominant integral highest weight Λ. Cylindric plane partitions actually parametrize a basis for V Λ ⊗ F , where F is the space spanned by partitions. We use this to calculate the partition function for a system of random cylindric plane partitions. We also observe a form of rank level duality. Finally, we use an explicit bijection to relate our work to the Kyoto path model.

Demazure crystals of generalized Verma modules and a flagged RSK correspondence

“…(3) Recently Tingley [33] gave a nice representation theoretic interpretation of cylindric plane partitions in terms of crystal graphs for affine Lie algebra sl n and its generating function. It would be interesting to find an application of affine Demazure crystals to cylindric plane partitions.…”

Quantum affine Gelfand–Tsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces

“…Restricting V (µ) to the subalgebra of Üv ( sl n ), generated by {e i,0 , f i,0 , v ±hi } 1≤i≤n which is isomorphic to U v ( sl n ) (called horizontal in [17]) we obtain the same named U v ( sl n )-module with the Gelfand-Tsetlin basis parameterized by D(µ). Recall that in the proof of Theorem 3.22, [4], there was constructed a bijection between D(µ) and Tingley's crystal B µ of cylindric plane partitions model of section 4 [15]. This answers Tingley's Question 1 ([15], p.38).…”

The $\mathrm{A}_2$ Andrews-Gordon identities and cylindric partitions

“…where |π| denotes the size of π. The first key observation, due to Foda and Welsh [22] (see also [64]), is that Borodin's product formula [8] for cylindric partitions implies that…”