Positive Jantzen sum formulas for cyclotomic Hecke algebras

Abstract

AbstractThis paper proves a “positive” Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type A and uses it to obtain new bounds on decomposition numbers. Quite remarkably, our results are proved entirely inside the cyclotomic Hecke algebras. Our positive sum formula shows that, in the Grothendieck group, the Jantzen sum formula can be written as an explicit non-negative linear combination of modules $$[E^{\varvec{\nu }}_{f,e}]$$ [ E f , e ν ] , which are the modular reductions of simple modules of related Hecke algebras in characteristic zero. The coefficient of $$[E^{\varvec{\nu }}_{f,e}]$$ [ E f , e ν ] in the sum formula is determined by the graded decomposition numbers in characteristic zero, which are known, and by the characteristic of the field. As a consequence we give an explicit upper bound for the decomposition numbers in characteristic $$p>0$$ p > 0 in terms of linear combinations of decomposition numbers of a cyclotomic Hecke algebra at $$ep^r$$ e p r th roots of unity in characteristic zero, for $$r\ge 0$$ r ≥ 0 . Finally, we prove a new and more elegant “classical” Jantzen sum formula for these algebras.