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A NOTE ON THE VARIETY OF PROJECTORS
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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 | ||||
| Official Date: | 10 September 1991 | ||||
| Dates: |
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| Volume: | 74 | ||||
| Number: | 1 | ||||
| Number of Pages: | 12 | ||||
| Page Range: | pp. 73-84 | ||||
| Publication Status: | Published |
Data sourced from Thomson Reuters' Web of Knowledge
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