Hilbert schemes, polygraphs and the Macdonald positivity conjecture

Author:

Haiman Mark

Abstract

We study the isospectral Hilbert scheme X n X_{n} , defined as the reduced fiber product of ( C 2 ) n (\mathbb {C}^{2})^{n} with the Hilbert scheme H n H_{n} of points in the plane C 2 \mathbb {C}^{2} , over the symmetric power S n C 2 = ( C 2 ) n / S n S^{n}\mathbb {C}^{2} = (\mathbb {C}^{2})^{n}/S_{n} . By a theorem of Fogarty, H n H_{n} is smooth. We prove that X n X_{n} is normal, Cohen-Macaulay and Gorenstein, and hence flat over H n H_{n} . We derive two important consequences. (1) We prove the strong form of the n ! n! conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients K λ μ ( q , t ) K_{\lambda \mu }(q,t) . This establishes the Macdonald positivity conjecture, namely that K λ μ ( q , t ) N [ q , t ] K_{\lambda \mu }(q,t)\in {\mathbb N} [q,t] . (2) We show that the Hilbert scheme H n H_{n} is isomorphic to the G G -Hilbert scheme ( C 2 ) n / / S n (\mathbb {C}^{2})^{n}{/\!\!/}S_n of Nakamura, in such a way that X n X_{n} is identified with the universal family over ( C 2 ) n / / S n ({\mathbb C}^2)^n{/\!\!/}S_n . From this point of view, K λ μ ( q , t ) K_{\lambda \mu }(q,t) describes the fiber of a character sheaf C λ C_{\lambda } at a torus-fixed point of ( C 2 ) n / / S n ({\mathbb C}^2)^n{/\!\!/}S_n corresponding to μ \mu . The proofs rely on a study of certain subspace arrangements Z ( n , l ) ( C 2 ) n + l Z(n,l)\subseteq (\mathbb {C}^{2})^{n+l} , called polygraphs, whose coordinate rings R ( n , l ) R(n,l) carry geometric information about X n X_{n} . The key result is that R ( n , l ) R(n,l) is a free module over the polynomial ring in one set of coordinates on ( C 2 ) n (\mathbb {C}^{2})^{n} . This is proven by an intricate inductive argument based on elementary commutative algebra.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Cited by 226 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3