On ideals of the group algebra of an infinite symmetric group over a field of characteristic \(p\). (Q1929774)

Let \(S_\infty\) be the infinite symmetric group of finite permutations and let \(FS_\infty\) be its group algebra over a field \(F\). When \(F\) is of characteristic 0, representation theory of finite symmetric groups \(S_n\) gives a description of the ideals of \(FS_\infty\) in the language of finite collections of partitions of integers. In the paper under review the author studies ideals of \(FS_\infty\) when \(F\) is a field of positive characteristic. The main result states that every nonzero ideal of \(FS_\infty\) contains skew-symmetric and symmetric elements of sufficiently large order, i.e., elements of the form \(\sum_{\sigma\in S_n}(-1)^\sigma\sigma\) and \(\sum_{\sigma\in S_n}\sigma\) for all sufficiently large \(n\). The proof is based on combinatorial techniques typical for the theory of algebras with polynomial identities. The exposition is reasonably self-contained. The only nontrivial fact given without proof is the deep theorem of the author [Int. J. Algebra Comput. 5, No. 2, 189-197 (1995; Zbl 0835.16019)] that in positive characteristic every PI-algebra satisfies some standard and some symmetric identity \[ \sum_{\sigma\in S_n}(-1)^\sigma x_{\sigma(1)}\cdots x_{\sigma(n)}=0\quad\text{ and }\quad\sum_{\sigma\in S_n}x_{\sigma(1)}\cdots x_{\sigma(n)}=0. \] Using the main result, the author transfers the question of the classification of the ideals of \(FS_\infty\), \(\text{char\,}F=p>0\), to the classification of certain subspaces of the tensor square of finitely generated free associative algebras.