Jantzen Filtration, Young Symmetrizers, and Young’s Seminormal Basis
For each partition of a positive integer , denote its associated Specht module of the symmetric group . This is a cyclic module generated by its Young symmetrizer , and has a distinguished basis called Young’s seminormal basis. It also has a well-known -Jantzen filtration.
Let be another partition, say of . We show that the -th term of the -Jantzen filtration of projects onto that of for all if the canonical projection splits over , the localised ring of at the prime ideal . Furthermore, this splitting condition can be explicitly stated in terms of the greatest common divisor of a certain product of Young symmetrizers, as well as in terms of the denominator of a certain Young’s seminormal basis vector.