Brauer Relations for Symmetric Groups: Young--Wreath Quotients and the n-Cycle Mark Obstruction

Abstract

Let $K(S_n)$ be the kernel of the linearization map from the Burnside ring of the symmetric group $S_n$ to its rational representation ring. We denote by $N_n$ the free $\mathbb{Z}$-sublattice with basis given by the standard Brauer relations $\Theta_H$, indexed by the $S_n$-conjugacy classes of subgroups that are either intransitive or transitive imprimitive and are not conjugate to Young subgroups. We also let $J_n$ be the $\mathbb{Z}$-lattice of relations induced from the kernels of the linearization maps for proper Young subgroups and standard wreath subgroups $S_a\wr S_b$, and put $O_n=N_n/J_n$. For every composite integer $n\geq 4$, we construct, using the $C_n$-mark and the Young section, an integral homomorphism $\omega_n:N_n\to\mathbb{Z}$ and prove directly over $\mathbb{Z}$ that $J_n=\ker(\omega_n|_{N_n})$ and $O_n\cong\mathbb{Z}$. We further construct a natural comparison homomorphism from $O_n$ to the primitive quotient $\operatorname{Prim}(S_n)$, obtained by factoring out all imprimitive relations arising from proper subquotients. Combined with the classification theorem of Bartel--Dokchitser, this shows that $O_n\xrightarrow{\sim}\operatorname{Prim}(S_n)$ for composite $n\geq 6$, while for $n=4$ the comparison map is reduction modulo $2$ under the identifications $O_4\cong\mathbb{Z}$ and $\operatorname{Prim}(S_4)\cong\mathbb{Z}/2\mathbb{Z}$. On the other hand, for prime $n\geq 5$, one has $O_n=0$, whereas $\operatorname{Prim}(S_n)\cong\mathbb{Z}$. Finally, we study the monomial Burnside ring over $C_2$ 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.