Kai Meng Tan, Ph.D.

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Jantzen Filtration, Young Symmetrizers, and Young’s Seminormal Basis

For each partition \lambda of a positive integer n, S^\lambda denote its associated Specht module of the symmetric group \mathfrak{S}^n. This is a cyclic module generated by its Young symmetrizer Y^\lambda, and has a distinguished basis called Young’s seminormal basis. It also has a well-known p-Jantzen filtration.

Let \mu be another partition, say of m. We show that the i-th term of the p-Jantzen filtration of S^{\lambda+\mu} projects onto that of S^\lambda for all i\in\mathbb{Z}^+ if the canonical projection \left(S^\lambda\boxtimes S^\mu\right)\uparrow_{\mathfrak{S}_n\times\mathfrak{S}_m}^{\mathfrak{S}_{n+m}}\twoheadrightarrow S^{\lambda+\mu} splits over \mathbb{Z}_{\left(p\right)}, the localised ring of \mathbb{Z} at the prime ideal \left(p\right). 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.

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