UNSPECIFIED. (1991) A NOTE ON THE VARIETY OF PROJECTORS. JOURNAL OF PURE AND APPLIED ALGEBRA, 74 (1). pp. 73-84. ISSN 0022-4049

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Abstract

Somewhat analogous to the case of the variety of Complexes, the variety P of projectors, i.e., idempotents (of rank d on an n-space), is shown to be a principal affine open subset of the product of the Grassmannian Gr(d, n) and its dual Gr(n - d, n). Also P is identified with the affine coset space GL(n)/H for a closed reductive subgroup H of the form GL(d) x GL(n - d); consequently, P is nonsingular and of dimension 2d(n - d). The coordinate ring R of P is described explicitly by generators and relations as the subring of left translation H-invariants of k[GL(n)] as an immediate consequence of the classical Hodge Standard Monomial Basis readily available for R just as for the homogeneous coordinate ring of Gr(d, n) for its Plucker embedding. The GL(n)-module structure of R is shown to be the direct limit of the filtered family of representations of GL(n): m-omega-d X m-omega-n-d X (-m) det., m element-of Z+, where omega-d and omega-n-d are the fundamental weights of GL(n) corresponding to Gr(d, n) and Gr(n - d, n), respectively, and det. is the determinant character of GL(n).

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Journal or Publication Title:	JOURNAL OF PURE AND APPLIED ALGEBRA
Publisher:	ELSEVIER SCIENCE BV
ISSN:	0022-4049
Date:	10 September 1991
Volume:	74
Number:	1
Number of Pages:	12
Page Range:	pp. 73-84
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/22447

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