Introduction to Quantum Groups and Crystal Bases

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

“…We use notation as in [6], and refer the reader to that book for a detailed explanation of crystals. For us, a crystal is a set B associated to a representation V of a symmetrizable Kac-Moody algebra g, along with operators e i : B → B ∪ {0} and f i : B → B ∪ {0}, which satisfy some conditions.…”

“…It turns out that wt(b) will always be equal to the weight of the corresponding canonical basis element v(b) (see for example [6]). …”

“…These in turn correspond to canonical basis vectors in V Λ (see, for example, [6] for a discussion of canonical bases). Let v(ψ) be the canonical basis element associated to γ(ψ).…”

“…We refer the reader to ( [6], Chapter 10) for the details of the Kyoto path model. We will only use the path models corresponding to one family B ℓ of perfect crystals for sl n , one of each level ℓ > 0.…”

“…al. in [10] and [11] (see [6] for a more recent explanation). We do this by exhibiting an explicit crystal isomorphism between the highest irreducible component of the abacus model and the Kyoto path model for a particular perfect crystal and ground state path.…”