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Another Combinatorial Determinant
We present a variation and generalization of a determinant evaluation of Wilf
(math.CO/9809120). His result concerns a matrix whose entries are the
coefficients of powers of a given power series; we replace the powers by
repeated compositions and obtain similar results.Comment: 3 pages, AMSLaTeX source (replaced 18 May 1998: corrected some typos,
added a reference provided by Herb Wilf
Invariant theory for singular -determinants
From the irreducible decompositions' point of view, the structure of the
cyclic -module generated by the -determinant degenerates when
. In this paper, we show that
-determinant shares similar properties which the ordinary determinant
possesses. From this fact, one can define a new (relative) invariant called a
wreath determinant. Using -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 -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 .Comment: 26 page
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