An explicit basis for the Grassmann \(T\)-space (Q308100)

Let \(K\langle X\rangle\) be the free associative algebra in countable set of variables \(X=\{x_1,x_2,\dots\}\). Let \(S^3\) denote the \(T\)-space generated by the commutator \([x_1,x_2,x_3]\), i.e., the subspace of \(K\langle X\rangle\) generated by images of this commutator under all endomorphisms of \(K\langle X\rangle\). Let \(P_n\subset K\langle X\rangle\) be the space of all multilinear polynomials of degree \(n\) in indeterminates \(\{x_1,\dots,x_n\}\). The authors construct an explicit linear basis for the \(S_n\)-module \(S^3\cap P_n\). See also [the authors, J. Algebra Appl. 7, No. 3, 319--336 (2008; Zbl 1163.16012)].