  We interpret a formula established by Lapid-M\'{\i}nguez on real regular
representations of ${\rm GL}_n$ over a local non-archimedean field as a matrix
determinant. We use the Lewis Carroll determinant identity to prove new
relations between real regular representations. Through quantum affine
Schur-Weyl duality, these relations generalize Mukhin-Young's Extended
$T$-systems, for representations of the quantum affine algebra
$U_q(\widehat{\mathfrak{sl}}_k)$, which are themselves generalizations of the
celebrated $T$-system relations.
