English

Abstract

Let V be the complex vector space of homogeneous linear polynomials in the variables x1,..., x m . Suppose G is a subgroup of S m , and χ is an irreducible character of G. Let H d (G, χ) be the symmetry class of polynomials of degree d with respect to G and χ. For any linear operator T acting on V, there is a (unique) induced operator K χ (T) ∈ End(H d (G, χ)) acting on symmetrized decomposable polynomials by $${K_\chi }\left( T \right)\left( {{f_1} * {f_2} * \cdots * {f_d}} \right) = T{f_1} * T{f_2} * \cdots * T{f_d}.$$ In this paper, we show that the representation T ↦ K χ (T) of the general linear group GL(V) is equivalent to the direct sum of χ(1) copies of a representation (not necessarily irreducible) T ↦ B χ G (T).