On admissible tensor products in $p$-adic Hodge theory

TL;DR

This paper investigates conditions under which tensor products and Schur functors of $B$-pairs in $p$-adic Hodge theory preserve certain admissibility properties, revealing the existence of specific characters that adjust these properties.

Contribution

It establishes new criteria for when tensor products and Schur functors of $B$-pairs are admissible, introducing the existence of characters to recover admissibility.

Findings

Existence of a character $ u$ such that $W( u^{-1})$ and $W'( u)$ are admissible when their tensor product is.

For large rank $W$, if a Schur functor applied to $W$ is admissible, then $W$ can be twisted to be admissible.

Results apply specifically to $p$-adic representations, extending the understanding of admissibility in $p$-adic Hodge theory.

Abstract

We prove that if $W$ and $W'$ are two $B$-pairs whose tensor product is crystalline (or semi-stable or de Rham or Hodge-Tate), then there exists a character $\mu$ such that $W(\mu^{-1})$ and $W'(\mu)$ are crystalline (or semi-stable or de Rham or Hodge-Tate). We also prove that if $W$ is a $B$-pair and $F$ is a Schur functor (for example $\Sym^n(-)$ or $\Lambda^n(-)$) such that $F(W)$ is crystalline (or semi-stable or de Rham or Hodge-Tate) and if the rank of $W$ is sufficiently large, then there is a character $\mu$ such that $W(\mu^{-1})$ is crystalline (or semi-stable or de Rham or Hodge-Tate). In particular, these results apply to $p$-adic representations.