A β\beta-ring is a commutative ring, RR, equipped with a set of operations, β H:RR\beta_H: R \to R, indexed by subgroups of symmetric groups, S nS_n, satisfying a number of conditions. They may be seen as collections of integral linear combinations of generalized symmetric powers defined on Burnside rings. The cohomotopy of a space, π 0(X)\pi^0(X), is a β\beta-ring (Guillot 06, Thrm 4.5).

They are completely unrelated to relational beta-modules.


Note that there are variations in the literature as to the definition of β\beta-rings. For a close comparison with λ-rings, see

Other references

  • Ernesto Vallejo, The free β\beta-ring on one generator, Journal of Pure and Applied Algebra 86(1), 1993, pp. 95-108, (doi)

  • N.W. Rymer, Power operations on the Burnside ring, J. London Math. Sot. (2) 15 (1977) 75-80.

  • E. Vallejo, Polynomial operations from Burnside rings to representation functors, J. Pure Appl. Algebra 65 (1990) 163-190.

  • G. Ochoa, Outer plethysm, Burnside rings and β\beta-rings, J. Pure Appl. Algebra 55 (1988), 173-195.

  • I. Morris and C.D. Wensley, Adams operations and λ-operations in β-rings, Discrete Mathematics Volume 50, 1984, Pages 253-270, (doi)

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