Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

EXPLICIT CAPELLI IDENTITIES FOR SKEW SYMMETRIC MATRICES

Kenji Kinoshitaa1 and Masato Wakayamaa2

a1 Atlas Information Service Co. Ltd, San Kitahama-biru 3-1-6 Kitahama, Chuo-ku, Osaka-shi, Osaka-fu, Japan 541-0041 (kinosita@atlas-is.co.jp)

a2 Department of Mathematics, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan (wakayama@math.kyushu-u.ac.jp)

Abstract

The main purpose of this paper is to obtain an explicit Capelli identity relating skew-symmetric matrices under the action of the general linear group $GL_N$. In particular, we give an explicit formula for the skew Capelli element in terms of the trace of powers of a matrix defined by the standard infinitesimal generators of $GL_N$.

AMS 2000 Mathematics subject classification: Primary 17B35. Secondary 17B10; 15A15; 20G05

(Received November 27 2000)

Keywords

  • Capelli identity;
  • skew Capelli element;
  • Pfaffian;
  • universal enveloping algebra;
  • invariant differential operator;
  • Young diagram