Two different approaches are used to construct infinite-component spinor equations based on the multiplicity-free irreducible representations of S̄L̄(4,R). These ‘‘manifield’’ equations are SL(2,C) invariant; they exist in special relativity, and can directly be coupled to gravitation in the metric-affine theory, i.e., in Einstein’s general relativity with nonpropagating torsion and nonmetricity. In the first approach the maximal compact subgroup S̄Ō(4) of S̄L̄(4,R) is ‘‘physical.’’ A vector operator X μ is constructed directly in the infinite-dimensional reducible representation 𝒟disc( 1/2 ,0) ⊕𝒟disc(0, 1/2 ). In the second approach, SL(2,C) and a vector operator γ μ are embedded directly in S̄L̄(4,R) via the Dirac representation. A manifield equation is then constructed (in a manner analogous to the Majorana equation) by taking an infinite-dimensional irreducible multiplicity-free representation of SL(4,R), spinorial in  j1, in the ( j1, j2) reduction over S̄Ō(4). Both manifields can fit the observed mass spectrum.

We consider the ^-component quantum Potts model on a ddimensional cubic lattice with symmetry breaking and transverse fields. The model is solved exactly in two special limiting cases: 1) the infinite latticedimensionality (d->oo) limit and 2) the limit of infinitely-weak, long-range interactions of Kac type. In each case the resulting free energy and its first partial derivatives (order parameters) are shown to be identical to the corresponding mean-field expressions.

Given a Lie group K acting on a principal fiber bundle P(M, G), we study the K-invariance of connections in P and in bundles associated to P. The geometry of spontaneous symmetry breaking is discussed from this point of view. The results are applied to the Wu–Yang and ’t Hooft–Polyakov models of a magnetic monopole.

A famous result of Hall asserts that the multiplication and exponentiation in finitely generated torsion-free nilpotent groups can be described by rational polynomials. We describe an algorithm to determine such polynomials for all torsion-free nilpotent groups of given Hirsch length. We apply this to determine the Hall polynomials for all such groups of Hirsch length at most 7.

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