Abstract
A simple formula is obtained by which any symmetrized power of D(j) may be expressed in terms of the symmetrized powers of D(j-1/2). This formula does not depend explicitly on the representation theory of the symmetric group and it gives a method of building up any symmetrized power in fully reduced form. In particular, recurrence formulae are obtained for the totally symmetrized and totally antisymmetrized powers of D(j), and a formula is given for arbitrary symmetrized powers of D(1). Finally it is proved that D(j) contains D(j-1)n, n>or=2, as a proper subrepresentation and formulae are obtained for the symmetrized cubes of D(j).
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