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Báo cáo toán học: "A Hessenberg generalization of the Garsia-Procesi basis for the cohomology ring of Springer varieties"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: A Hessenberg generalization of the Garsia-Procesi basis for the cohomology ring of Springer varieties. | A Hessenberg generalization of the Garsia-Procesi basis for the cohomology ring of Springer varieties Aba Mbirika Department of Mathematics Bowdoin College Brunswick Maine USA ambirika@bowdoin.edu www.bowdoin.edu ambirika Submitted Jan 7 2010 Accepted Oct 29 2010 Published Nov 11 2010 Mathematics Subject Classifications 05E15 014M15 Abstract The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976 T.A. Springer observed that this variety s cohomology ring carries a symmetric group action and he offered a deep geometric construction of this action. Sixteen years later Garsia and Procesi made Springer s work more transparent and accessible by presenting the cohomology ring as a graded quotient of a polynomial ring. They combinatorially describe an explicit basis for this quotient. The goal of this paper is to generalize their work. Our main result deepens their analysis of Springer varieties and extends it to a family of varieties called Hessenberg varieties a two-parameter generalization of Springer varieties. Little is known about their cohomology. For the class of regular nilpotent Hessenberg varieties we conjecture a quotient presentation for the cohomology ring and exhibit an explicit basis. Tantalizing new evidence supports our conjecture for a subclass of regular nilpotent varieties called Peterson varieties. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R153 1 Contents 1 Introduction 2 1.1 Brief history of the Springer setting. 4 1.2 Definition of a Hessenberg variety. 5 1.3 Using h b -fillings to compute the Betti numbers of Hessenberg varieties . 5 1.4 The map from h b -fillings to monomials Ah b . 6 2 The Springer setting 7 2.1 Remarks on the map when h 1 2 . n . 8 2.2 The inverse map T from monomials in A b to h b -fillings. 8 2.3 A b coincides with the Garsia-Procesi basis B b . 12 3 The regular nilpotent Hessenberg setting 16 3.1 The ideal Jh the quotient ring R Jh and its basis Bh b . 17 3.2 Constructing an h-tableau-tree.