  The global symmetry algebras of partially-massless (PM) higher-spin (HS)
fields in (A)dS$_{d+1}$ are studied. The algebras involving PM generators up to
depth $2\,(\ell-1)$ are defined as the maximal symmetries of free conformal
scalar field with $2\,\ell$ order wave equation in $d$ dimensions. We review
the construction of these algebras by quotienting certain ideals in the
universal enveloping algebra of $(A)dS_{d+1}$ isometries. We discuss another
description in terms of Howe duality and derive the formula for computing trace
in these algebras. This enables us to explicitly calculate the bilinear form
for this one-parameter family of algebras. In particular, the bilinear form
shows the appearance of additional ideal for any non-negative integer values of
$\ell-d/2\,$, which coincides with the annihilator of the one-row $\ell$-box
Young diagram representation of $\mathfrak{so}_{d+2}\,$. Hence, the
corresponding finite-dimensional coset algebra spanned by massless and PM
generators is equivalent to the symmetries of this representation.
