Brundan, Kleshchev and Wang endow the Specht modules S λ over the cyclotomic Khovanov-Lauda-Rouquier algebra H Λ n with a homogeneous Z-graded basis. In this paper, we begin the study of graded Specht modules labelled by hook bipartitions ((n−m), (1 m )) in level 2 of H Λ n , which are precisely the Hecke algebras of type B, with quantum characteristic at least three. We give an explicit description of the action of the Khovanov-Lauda-Rouquier algebra generators ψ 1 , .
This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these results and other modular representation theoretic techniques on the study of Kronecker coefficients and hence verify Saxl’s conjecture for several large new families of partitions. In particular, we verify Saxl’s conjecture for all irreducible characters of
S
n
\mathfrak {S}_n
which are of 2-height zero.
We continue the study of Specht modules labelled by hook bipartitions for the Iwahori-Hecke algebra of type B with e ∈ {3, 4, . . . } via the cyclotomic Khovanov-Lauda-Rouquier algebra H Λ n . Over an arbitrary field, we explicitly determine the graded decomposition submatrices for H Λ n comprising rows corresponding to hook bipartitions.
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