On the Center Conjecture for the Cyclotomic KLR Algebras

The center conjecture for the cyclotomic KLR algebras R_β^∧ asserts that the center of R_β^∧ consists of symmetric elements in its KLR x and e(ν) generators. In this paper, we show that this conjecture is equivalent to the injectivity of some natural map ι^∧_β^,i from the cocenter of R_β^∧ to the cocenter of R_β^∧+^∧i for all i ∈ I and ∧ ∈ P⁺. We prove that the map ι^∧_β^,i is given by multiplication with a center element z(i, β) ∈ R_β^∧+^∧i and we explicitly calculate the element z(i, β) in terms of the KLR x and e(ν) generators. We present explicit monomial bases for certain bi-weight spaces of the defining ideal of R_β^∧.