Generalised column removal for graded homomorphisms between Specht modules
Let n be a positive integer, and let $$\mathscr {H}_n$$Hn denote the affine KLR algebra in type A. Kleshchev, Mathas and Ram have given a homogeneous presentation for graded column Specht modules $${\text {S}}_{\lambda }$$Sλ for $$\mathscr {H}_n$$Hn. Given two multipartitions $$\lambda $$λ and $$\mu $$μ, we define the notion of a dominated homomorphism $${\text {S}}_{\lambda }\rightarrow {\text {S}}_{\mu }$$Sλ→Sμ and use the KMR presentation to prove a generalised column removal theorem for graded dominated homomorphisms between Specht modules. In the process, we prove some useful properties of $$\mathscr {H}_n$$Hn-homomorphisms between Specht modules which lead to an immediate corollary that, subject to a few demonstrably necessary conditions, every homomorphism $${\text {S}}_{\lambda }\rightarrow {\text {S}}_{\mu }$$Sλ→Sμ is dominated, and in particular $${{\mathrm{Hom}}}_{\mathscr {H}_n}({\text {S}}_{\lambda },{\text {S}}_{\mu })=0$$HomHn(Sλ,Sμ)=0 unless $$\lambda $$λ dominates $$\mu $$μ. Brundan and Kleshchev show that certain cyclotomic quotients of $$\mathscr {H}_n$$Hn are isomorphic to (degenerate) cyclotomic Hecke algebras of type A. Via this isomorphism, our results can be seen as a broad generalisation of the column removal results of Fayers and Lyle and of Lyle and Mathas, generalising both into arbitrary level and into the graded setting.