  From the irreducible decompositions' point of view, the structure of the
cyclic $GL_n$-module generated by the $\alpha$-determinant degenerates when
$\alpha=\pm \frac1k (1\leq k\leq n-1)$. In this paper, we show that
$-\frac1k$-determinant shares similar properties which the ordinary determinant
possesses. From this fact, one can define a new (relative) invariant called a
wreath determinant. Using $(GL_m, GL_n)$-duality in the sense of Howe, we
obtain an expression of a wreath determinant by a certain linear combination of
the corresponding ordinary minor determinants labeled by suitable rectangular
shape tableaux. Also we study a wreath determinant analogue of the Vandermonde
determinant, and then, investigate symmetric functions such as Schur functions
in the framework of wreath determinants. Moreover, we examine coefficients
which we call $(n,k)$-sign appeared at the linear expression of the wreath
determinant in relation with a zonal spherical function of a Young subgroup of
the symmetric group $S_{nk}$.
