Affine Hecke and Schur algebras of type A without a square root of q

We provide an affine cellular structure on the extended affine Hecke algebra and affine $q$-Schur algebra of type $A_{n-1}$ that is defined over $\mathbb{Z}\left[q^{\pm1}\right]$, that is, without an adjoined $q^{\frac{1}{2}}$. This is with an eye to applications in the representation theory of $\mathrm{GL}_n(F)$ for a $p$-adic field $F$ over coefficient rings in which $p$ is invertible but does not have a square root, which have been a topic of recent interest. This is achieved via a renormalisation of the known affine cellular structure over $\mathbb{Z}\left[q^{\pm\frac{1}{2}}\right]$ at each left and right cell, which is chosen to ensure that the diagonal intersections remain subalgebras and that the left and right cells remain isomorphic. We furthermore show that the affine cellular structure on the Schur algebra has idempotence properties which imply finite global dimension, an important ingredient for the applications to representations of $p$-adic groups.