Representations of Quantum Algebras and Combinatorics of Young Tableaux

A diagrammatic approach to categorification of quantum groups I

“…The work of Ariki [1] can be viewed as a categorification of the restricted dual of U − (g) for g = sl N and g = sl N and a categorification of all irreducible integrable representations of these Lie algebras (see also [27], [2], [3], [37]). An integral version of the restricted dual of U − (g) becomes the sum of Grothendieck groups of suitable blocks of affine Hecke algebra representations.…”

Dipper–James–Murphy’s Conjecture for Hecke Algebras of Type B n

“…Unlike the case of the BGG category, we may have D λ = 0 and it is important to know when it occurs. When −Q is not a power of q, a bipartition λ = (λ (1) , λ (2) ) is (Q, e)-restricted if and only if both λ (1) and λ (2) are e-restricted. Thus we know when a bipartition is (Q, e)-restricted.…”

“…More precise definition is given in the next section and λ = (λ (1) , λ (2) ) is Kleshchev if and only if λ (2) ⊗λ (1) belongs to the subcrystal B(Λ 0 +Λ m ) of B(Λ 0 )⊗B(Λ m ), where the crystals B(Λ 0 ) and B(Λ m ) are realized on the set of e-restricted partitions. Now, D λ = 0 if and only if λ is Kleshchev by [3].…”

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

“…This formula for calculating e i and f i follows immediately from Equations (1) and (2). To see that e i (a ⊗ b) = 0 if there are any uncanceled ")", notice that in this case you always act on a factor that contributes at least one ")", and hence has ǫ i > 0.…”