## University of Sydney Algebra Seminar

# Jun Hu (University of Sydney)

## Friday 1st May, 12.05-12.55pm, Carslaw 375

### On a generalisation of the Dipper--James--Murphy Conjecture

Let e be 0 or an integer bigger than 1. Let v_1,...,v_r \in Z/eZ. The notion of Kleshchev r-multipartitions with respect to (e;v_1,...,v_r) plays an important role in the modular representation theory of the cyclotomic Hecke algebras H_n(q;q^{v_1},...,q^{v_r}) as well as the crystal bases theory of integrable highest weight modules over the affine Lie algebras of type A_{e-1}^{(1)} if e>1; or of type A_{\infty} if e=0. In the case r=1, they are the same as e-restricted partitions. In the case r=2, Dipper, James and Murphy proposed in 1995 a conjecture which asserts that Kleshchev bipartitions are the same as (Q,e)-restricted bipartitions. This conjecture was recently solved by Ariki--Jacon. In this talk, I will explain a natural generalisation of the Dipper--James--Murphy conjecture to the case r>2 and my proof of this conjecture in the multi-core case. In particular, if e=0, the generalised Dipper--James--Murphy Conjecture is always true for any integer r.