Crystal duality and Littlewood–Richardson rule of extremal weight crystals

Abstract

We consider a category of gl ∞ -crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is anti-isomorphic to an Ore extension of the character ring of integrable lowest weight gl ∞ -modules with respect to derivations shifting the characters of fundamental weight modules. A Littlewood-Richardson rule of extremal weight crystals of non-negative level is described explicitly in terms of classical Littlewood-Richardson coefficients.

Crystal bases of modified quantized enveloping algebras and a double RSK correspondence

“…Suppose that A is J-admissible and l-admissible for some l ∈ I • . Then both A and X l A generate the same J-colored oriented graphs with respect to e k and f k for k ∈ J • whenever X l A = 0 (X = E, F ) [13,Lemma 3.2]. A similar fact holds when A is I-admissible and k-admissible for some k ∈ J • .…”

“…Note that for µ ∈ P, B µ has neither highest nor lowest weight vector. It is shown in [13] that for µ, ν ∈ P, Let Z n + = { λ = (λ 1 , . .…”

“…Note that {B µ,ν ⊗ B(Λ) | Λ ∈ P + , µ, ν ∈ P } forms a complete list of extremal weight crystals of non-negative level up to isomorphism [13,Proposition 3.12].…”

“…m+n],Z .By[13, Proposition 4.5], F m ⊗ (F n ) ∨ is a disjoint union of extremal weight gl ∞crystals of level m − n, and hence so is B(Λ µ ) ⊗ B(−Λ ν ).For r ∈ Z, we define B >r (µ, ν) to be the set of A = (a ij ) ∈ M [m+n],Z such that…”

Tensor product multiplicities for crystal bases of extremal weight modules over quantum infinite rank affine algebras of types $B_{∞}$, $C_{∞}$, and $D_{∞}$

“…Let U q (g) be the quantized universal enveloping algebra over C(q) associated to the infinite rank affine Lie algebra g of type A +∞ , A ∞ , B ∞ , C ∞ , or D ∞ with Cartan subalgebra h = i∈I Ch i and integral weight lattice P = i∈I ZΛ i ⊂ h * , where I is the (infinite) index set for the simple roots. In [Kw2], [Kw3], Kwon studied the crystal basis B(λ) of the extremal weight U q (g)-module V (λ) of extremal weight λ ∈ P in the cases where g is of type A +∞ and type A ∞ . In these papers, he gave a combinatorial realization of the crystal basis B(λ) for λ ∈ P of level zero (see also [Kw1,§4.1] for the case of dominant λ ∈ P ) by using semistandard Young tableaux in which the entries are the crystal basis elements of the vector representation of U q (g).…”