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We study the circular polarization of the Mn I resonance lines at 279.56, 279.91, and 280.19 nm (hereafter, UV multiplet) by means of radiative transfer modeling. In 2019, the CLASP2 mission obtained unprecedented spectropolarimetric data in a region of the solar ultraviolet including the Mg II h and k resonance lines and two lines of a subordinate triplet, as well as two Mn I resonance lines. The first analysis of such data, in particular those corresponding to a plage region, allowed the inference of the longitudinal magnetic field from the photosphere to the upper chromosphere just below the transition region. This was achieved by applying the weak field approximation to the circular polarization profiles of the Mg II and Mn I lines. While the applicability of this approximation to the Mg II lines was already demonstrated in previous works, this is not the case for the Mn I UV multiplet. These lines are observed as absorptions between the Mg II h and k lines, a region whose intensity is shaped by their partial frequency redistribution effects. Moreover, the only Mn I stable isotope has nuclear spin $I=5/2$ and thus hyperfine structure must be, a priori, taken into account. Here we study the generation and transfer of the intensity and circular polarization of the Mn I resonance lines accounting for these physical ingredients. We analyze their sensitivity to the magnetic field by means of their response function, and we demonstrate the applicability of the weak field approximation to determine the longitudinal component of the magnetic field.