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For instance, the unitary group $U(1)$ is useful in quantum physics, the special orthogonal group $SO(3,1)$ is useful in special relativity, $SU(2)$ for electroweak force and $SU(3)$ for chromodynamics, etc.

Is there a use for $GL(4,R)$ or $GL^+(4,R)$ as a fundamental group in physics?

Anon21
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  • It's unclear what "fundamental application" is supposed to mean, but you may find https://physics.stackexchange.com/q/225413 useful – Nihar Karve Jan 24 '21 at 04:31
  • $GL(n,\mathbb{R})$ is used in almost all areas of physics and mathematics, e.g. linear algebra. – Qmechanic Jan 24 '21 at 06:34
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    @Qmechanic You are sure about that? I spend 2 weeks googling papers and almost everything I have seen takes at most sub-group of the general linear group. The only exception -in the sense that it is a "super-group" of such- is metric-affine general relativity, but even it takes the $T^4 \times GL(4,R)$ and not just $GL(4,R)$ - not to mention that metric-affine GR is not proven experimentally and may thus only be a mathematical novelty. But as far as using GL(4,R) by itself for an actual law of physics... nothing! – Anon21 Jan 25 '21 at 00:40

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