# The Library

### 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

**Full text not available from this repository.**

## 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 |

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 |

Data sourced from Thomson Reuters' Web of Knowledge

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