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

We prove that if $W$ and $W^{\mathrm{\prime }}$ 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 $W^{\mathrm{\prime }}(\mu )$ 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 ${\text{Sym}}^n(-)$ or ${\mathrm{\Lambda }}^n(-)$) 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.