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We interpret a formula established by Lapid-M\'{\i}nguez on real regularrepresentations of ${\rm GL}_n$ over a local non-archimedean field as a matrixdeterminant. We use the Lewis Carroll determinant identity to prove newrelations between real regular representations. Through quantum affineSchur-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 thecelebrated $T$-system relations.

On a determinant formula for some real regular representations