2006/03/31 by Kazufumi Kimoto, Masato Wakayama · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #math.RT #msc:15A72 #msc:17B10
paper · pdf · doi:10.1016/j.jcta.2007.03.008
published as J. Combin. Theory Ser. A 115 (2008), no.1, 1--31 · 26 pages
arxiv created 2007/02/28 · openalex publication_date 2007/04/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
From the irreducible decompositions' point of view, the structure of the cyclic GLn-module generated by the α-determinant degenerates when α=± \frac1k (1≤ k≤ 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 (GLm, GLn)-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 Snk.