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The theory of correlated electron systems on a lattice proves notoriouslycomplicated because of the exponential growth of Hilbert space. Mean-fieldapproaches provide valuable insight when the self-energy has a dominant localstructure. Additionally, the extraction of effective low-energy theories fromthe generalized many-body representation is highly desirable. In this respect,the rotational-invariant slave boson (RISB) approach in its mean-fieldformulation enables versatile access to correlated lattice problems. However inits original form, due to numerical complexity, the RISB approach is limited toabout three correlated orbitals per lattice site. We thus present a thoroughsymmetry-adapted advancement of RISB theory, suited to efficiently deal withmulti-orbital Hubbard Hamiltonians for complete atomic-shell manifolds. It isutilized to study the intriguing problem of Hund's physics for three- andespecially five-orbital manifolds on the correlated lattice, includingcrystal-field terms as well as spin-orbit interaction. The well-knownJanus-face phenomenology, i.e. strenghtening of correlations atsmaller-to-intermediate Hubbard $U$ accompanied by a shift of the Motttransition to a larger $U$ value, has a stronger signature and more involvedmultiplet resolution for five-orbital problems. Spin-orbit interactioneffectively reduces the critical local interaction strength and weakens theJanus-face behavior. Application to the realistic challenge of Fe chalcogenidesunderlines the subtle interplay of the orbital degrees of freedom in thesematerials

Rigorous symmetry adaptation of multiorbital rotationally invariant slave-boson theory with application to Hund's rules physics