Asymptotic results on modular representations of symmetric groups and almost simple modular group algebras
This paper proves two lovely results. To start with, let \(\text{Alt}(\Omega)\) denote the alternating group on the finite set \(\Omega\). An embedding \(\text{Alt}(\Omega)\to\text{Alt}(\Lambda)\) is called diagonal if all orbits of \(\text{Alt}(\Omega)\) on \(\Lambda\) have length 1 or \(|\Omega|\). If \(\text{Alt}(\Omega_1)\to\text{Alt}(\Omega_2)\to\text{Alt}(\Omega_3)\to\cdots\) is an infinite sequence of proper embeddings, then the locally finite simple group \(G=\bigcup_{i=1}^\infty\text{Alt}(\Omega_i)\) is said to be a limit alternating group, and it is diagonal if all but finitely many of the embeddings \(\text{Alt}(\Omega_i)\to\text{Alt}(\Omega_{i+1})\) are diagonal. Otherwise, \(G\) is non-diagonal. The first main result is: Theorem A. Let \(F\) be a field of characteristic \(\neq 2\) and let \(G\) be a limit alternating group. Then the group algebra \(F[G]\) is almost simple if and only if \(G\) is non-diagonal. Note that, for any group \(G\), \(F[G]\) is almost simple if and only if its only proper ideal is the kernel of the augmentation homomorphism \(F[G]\to F[G/G]=F\). A key ingredient in the proof of the above is a new asymptotic theorem on the representations of the symmetric group \(\Sigma_n=\text{Sym}(n)\) in characteristic \(p>0\). Here \(\Sigma_n\subseteq\Sigma_i\) indicates the natural embedding of one symmetric group in another. One aspect of this result is: Theorem B. Let \(F\) be an algebraically closed field of characteristic \(p>3\), and suppose that \(n>(p-1)^2\). Then there exists an integer \(N>n\) such that, for any \(i\geq N\), the restriction of any faithful \(F[\Sigma_i]\)-module to \(\Sigma_n\) has, as a composition factor, either the natural irreducible module \(V_n\) for \(\Sigma_n\), or the module \(V_n\otimes(\text{sgn}_n)\), or a faithful completely splittable \(\Sigma_n\)-module. Note that the irreducible modules of \(F[\Sigma_n]\) are characterized by the \(p\)-regular partitions of \(n\), and such a module is said to be completely splittable if its corresponding partition satisfies a certain technical condition.
scientific article; zbMATH DE number 1346317