A proof of the Mullineux conjecture
Let \(\lambda=(\nu^{\alpha_1}_1,\dots,\nu^{\alpha_k}_k)\), \(\nu_1>\cdots>\nu_k>0\), \(\alpha_i>0\), be a partition of \(n\in\mathbb{N}\). \(\lambda\) is said to be \(p\)-regular for a prime \(p\) if \(\alpha_1,\dots,\alpha_k<p\). Let \(F\) be a field of characteristic \(p>0\). All irreducible representations of the symmetric group \(S_n\) are naturally parametrized by \(p\)-regular partitions. We denote by \(D^\lambda\) the irreducible module corresponding to \(\lambda\). Let \(\text{sgn}_n(g)=\text{sgn}(g)\) be the sign of a permutation \(g\). Then, given an irreducible \(D^\lambda\), the tensor product \(D^\lambda\otimes\text{sgn}_n\) is also irreducible. Put \(D^\lambda\otimes\text{sgn}_n=D^{b_n(\lambda)}\). In this way a bijection \(b_n\) on the set \(P_n\) of \(p\)-regular partitions of \(n\) is defined for each \(n\in\mathbb{N}\). Mullineux proposed an algorithm which defines a bijection \(m_n\colon P_n\to P_n\) and conjectured that \(b_n=m_n\). The authors present a proof of this conjecture.
scientific article; zbMATH DE number 1096019
scientific article