Decomposable critical tensors

Let λ = (λ1, … , λt) be a partition of m and its conjugate partition. Denote also by λ the irreducible C-character of Sm associated with λ. Let V be a finite dimensional vector space over C.The reach of an element of the symmetry class of tensors Vλ (symmetry class of tensors associated with λ) is defined. The concept of critical element is introduced, as an element whose reach has dimension equal to . It is observed that, in ∧mV, the notions of critical element and decomposable element coincide. Known results for decomposable elements of ∧mV are extended to critical elements of Vλ. In particular, for a basis of ⊗mV induced by a basis of V, generalized Plücker polynomials are constructed in a way that the set of their common roots contains the set of the families of components of decomposable critical elements of Vλ.