Invariant theory for singular $α$-determinants
| dc.creator | Kimoto, Kazufumi | |
| dc.creator | Wakayama, Masato | |
| dc.date | 2006-03-30 | |
| dc.date | 2007-02-28 | |
| dc.date.accessioned | 2026-07-07T08:43:49Z | |
| dc.date.available | 2026-07-07T08:43:49Z | |
| dc.description | From the irreducible decompositions' point of view, the structure of the cyclic $GL_n$-module generated by the $α$-determinant degenerates when $α=\pm \frac1k (1\leq k\leq n-1)$. 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 $S_{nk}$. | |
| dc.description | 26 pages | |
| dc.identifier | https://arxiv.org/abs/math/0603699 | |
| dc.identifier | http://arxiv.org/abs/math/0603699 | |
| dc.identifier | J. Combin. Theory Ser. A 115 (2008), no.1, 1--31 | |
| dc.identifier | doi:10.1016/j.jcta.2007.03.008 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/142452 | |
| dc.subject | Representation Theory | |
| dc.subject | 17B10; 15A72 | |
| dc.title | Invariant theory for singular $α$-determinants | |
| dc.type | text |