On graded decomposition numbers for cyclotomic Hecke algebras in quantum characteristic 2

Brundan and Kleshchev introduced graded decomposition numbers for representations of cyclotomic Hecke algebras of type A, which include group algebras of symmetric groups. Graded decomposition numbers are certain Laurent polynomials, whose values at 1 are the usual decomposition numbers. We show that in quantum characteristic 2 every such polynomial has non-zero coefficients either only in odd or only in even degrees. As a consequence, we find the first examples of graded decomposition numbers of symmetric groups with non-zero coefficients in some negative degrees.