2026/07/27 by Wakatake Masahiro
Mathematics · #math.GR #math.RT
Let K(Sn) be the kernel of the linearization map from the Burnside ring of the symmetric group Sn to its rational representation ring. We denote by Nn the free ℤ-sublattice with basis given by the standard Brauer relations ΘH, indexed by the Sn-conjugacy classes of subgroups that are either intransitive or transitive imprimitive and are not conjugate to Young subgroups. We also let Jn be the ℤ-lattice of relations induced from the kernels of the linearization maps for proper Young subgroups and standard wreath subgroups Sa\wr Sb, and put On=Nn/Jn. For every composite integer n≥ 4, we construct, using the Cn-mark and the Young section, an integral homomorphism ωn:Nn→ℤ and prove directly over ℤ that Jn=ker(ωn|Nn) and On≅ℤ. We further construct a natural comparison homomorphism from On to the primitive quotient Prim(Sn), obtained by factoring out all imprimitive relations arising from proper subquotients. Combined with the classification theorem of Bartel--Dokchitser, this shows that On\xrightarrow∼Prim(Sn) for composite n≥ 6, while for n=4 the comparison map is reduction modulo 2 under the identifications O4≅ℤ and Prim(S4)≅ℤ/2ℤ. On the other hand, for prime n≥ 5, one has On=0, whereas Prim(Sn)≅ℤ. Finally, we study the monomial Burnside ring over C2 and show, via an additive map that we call index-two permutation reduction, that the resulting quotient is canonically isomorphic to the ordinary Young--wreath quotient.