Symmetrizations of tensors by irreducible characters of the symmetric group serve as natural analogues of symmetric and skew-symmetric tensors. The question of when a symmetrized decomposable tensor is non-zero is intimately related to the rank partition of a matroid extracted from the tensor. In this paper we characterize the non-vanishing of the symmetrization of certain partially symmetrized...

Studied is an assumption on a group that ensures that no matter how the group is embedded in a symmetric group, the corresponding symmetrized tensor space has an orthogonal basis of standard (decomposable) symmetrized tensors.

‎in this paper‎, ‎we give a necessary and sufficient condition for the equality of two symmetrized decomposable polynomials‎. ‎then‎, ‎we study some algebraic and geometric properties of the induced operators over symmetry classes of polynomials in the case of linear characters‎.

It is shown that the full symmetry class of tensors associated with the irreducible character [2, 1n−2] of Sn does not have an orthogonal basis consisting of decomposable symmetrized tensors.

The Macaulay2 package DecomposableSparseSystems implements methods for studying and numerically solving decomposable sparse polynomial systems. We describe the structure of systems explain how in this may be used to exploit structure, with examples.