Positive Jantzen sum formulas for cyclotomic Hecke algebras
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 nonnegative 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 $r\ge0$.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.15486
 Bibcode:
 2021arXiv210615486M
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Combinatorics;
 Mathematics  Quantum Algebra;
 20G43;
 20C08;
 20C30;
 05E10
 EPrint:
 LaTeX, 32 pages, TikZ diagrams and LaTeX3 tables. Graded adjustment matrix counterexample added to Remark 4.22. Comments welcome