Abstract
We provide two models for the free β-ring on one generator. One is the graded ring S( A) = ⊕ ∞ n=0 AS( n), where AS( n) denotes the Burnside ring of the symmetric group S( n) of degree n. This gives an answer to a question raised by Ochoa in [Outer plethysm, Burnside ring and β-rings, J. Pure Appl. Algebra 55 (1988) 173–195]. The second model is the ring of geometric operators on the Burnside ring functor introduced by the author in [Polynomial operations from Burnside rings to representation functors, J. Pure Appl. Algebra 65 (1990) 163–190].
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