Positive Jantzen sum formulas for cyclotomic Hecke algebras

TL;DR

This paper establishes a positive Jantzen sum formula for cyclotomic Hecke algebras of type A, linking modular decomposition numbers across different roots of unity and characteristics, enhancing understanding of their representation theory.

Contribution

It introduces a positive sum formula for Specht modules in cyclotomic Hecke algebras, connecting modular decomposition numbers to characteristic zero data and different roots of unity.

Findings

Sum of Jantzen filtration pieces equals a non-negative combination of modules

Decomposition numbers depend on related algebras at higher roots of unity

Explicit formula involves graded decomposition numbers and field characteristic

Abstract

We prove a ``positive'' Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type~$A$. That is, in the Grothendieck group, we show that the sum of the pieces of the Jantzen filtration is equal to an explicit non-negative linear combination of modules $E^\nu_{f,e}$, which are modular reductions of simple modules for closely connected Hecke algebras in characteristic zero. The coefficient of $E^\nu_{f,e}$ in the sum formula is determined by the graded decomposition numbers in characteristic zero, which are known, and the characteristic of the field. As a consequence we see that the decomposition numbers of a cyclotomic Hecke algebra at an $e$th root of unity in characteristic $p$ depend on the decomposition numbers of related cyclotomic Hecke algebras at $ep^r$th roots of unity in characteristic zero, for $r\ge0$.