  We formalize some known categorical equivalences to give a rigorous treatment
of smooth representations of p-adic general linear groups, as ungraded modules
over quiver Hecke algebras of type A.
  Graded variants of RSK-standard modules are constructed for quiver Hecke
algebras. Exporting recent results from the p-adic setting, we describe an
effective method for construction and classification of all simple modules as
quotients of modules induced from maximal homogenous data.
  It is established that the products involved in the RSK construction fit the
Kashiwara-Kim notion of normal sequences of real modules. We deduce that
RSK-standard modules have simple heads, devise a formula for the shift of
grading between RSK-standard and simple self-dual modules, and establish
properties of their decomposition matrix, thus confirming expectations for
p-adic groups raised in a work of the author with Lapid.
  Subsequent work will exhibit how the presently introduced RSK construction
generalizes the much-studied Specht construction, when inflated from cyclotomic
quotient algebras.
