Modular group rings of the finitary symmetric group (Q677436)

This paper offers a brief survey of some methods used to construct two-sided ideals in the group algebra \(K[S]\) of the finitary symmetric group \(S=\text{FSym}_\infty=\bigcup_{n=1}^\infty S_n\), where \(S_n=\text{Sym}_n\). The first section considers the annihilators of induced modules and states the necessary and sufficient conditions due to \textit{S. K. Sehgal} and the author [Nova J. Algebra Geom. 2, No. 1, 89-105 (1993; Zbl 0874.20004)] for the annihilator of a permutation module \((1_H)^S\) to be nonzero. Next, for any integer \(k\), let \(I(n,k)\) denote the annihilator in \(K[S_n]\) of its natural action on the \(K\)-linear span of all monomials of degree \(n\) in the free algebra \(K\langle X_k\rangle\) on the set \(X_k=\{x_1,x_2,\ldots,x_k\}\). Since \(I(n,k)\subseteq I(n+1,k)\), it follows that \(I(k)=\bigcup_{n=1}^\infty I(n,k)\) is an ideal of \(K[S]\). The second section of this paper discusses the result of \textit{E. Formanek} and \textit{C. Procesi} [Adv. Math. 19, 292-305 (1976; Zbl 0346.20021)] which asserts that the factor ring \(K[S]/I(k)\) has no nonzero nil ideals. It also contains a proof that \(I(k)\) is the annihilator of an induced module. The remainder of this survey is concerned with the work of \textit{Yu. P. Razmyslov} as described in his book [Identities of algebras and their representations (Nauka, Moscow 1989; Zbl 0673.17001)]. Specifically, let \(\theta\colon K[S_n]\to K[t]=T\) be the \(K\)-linear map given by \(\theta(g)=t^{c_n(g)}\), where \(c_n(g)\) is one less than the number of cycles (including those of length 1) of the permutation \(g\in S_n\). Furthermore, if \(I\) is any ideal of the polynomial ring \(T\), let \(L_n(I)\) be the set of all \(\alpha\in K[S_n]\) with \(\theta(\alpha h)\in I\) for all \(h\in S_n\). Then \(L_n(I)\) is an ideal of \(K[S_n]\) and \(L_n(I)\subseteq L_{n+1}(I)\). Hence \(L(I)=\bigcup_{n=1}^\infty L_n(I)\) is an ideal of \(K[S]\) which turns out to be prime if \(I\) is a prime ideal of \(T\) different from \(tT\). The final section of this paper goes on to describe the ideals \(L_n(I)\) and \(L(I)\) in terms of Young diagrams when \(K\) is a field of characteristic 0.