Total positivity, Schubert positivity, and geometric Satake

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• 作者：Thomas Lam^a ; ¹ ; ^{tfylam@umich.edu" class="auth_mail" title="E-mail the corresponding author} ; Konstanze Rietsch^b ; ² ; ^{konstanze.rietsch@kcl.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：20G20 ; 15A45 ; 14N35 ; 14N15
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 August 2016
• 年：2016
• 卷：460
• 期：Complete
• 页码：284-319
• 全文大小：684 K

文摘

Let G   be a simple, simply-connected complex algebraic group, and let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300564&_mathId=si1.gif&_user=111111111&_pii=S0021869316300564&_rdoc=1&_issn=00218693&md5=1442fe2f5a313f545cd10198d28ae5e7" title="Click to view the MathML source">X⊂G^∨class="mathContainer hidden">class="mathCode">$X\subset G^{\vee }$ be the centralizer of a principal nilpotent. Ginzburg and Peterson independently related the ring of functions on X   with the homology ring of the affine Grassmannian class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300564&_mathId=si188.gif&_user=111111111&_pii=S0021869316300564&_rdoc=1&_issn=00218693&md5=e07d47ea6f2e9e7584febb43af765ae4" title="Click to view the MathML source">Gr_Gclass="mathContainer hidden">class="mathCode">${\mathrm{Gr}}_G$. Peterson furthermore connected X   to the quantum cohomology rings of partial flag varieties class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300564&_mathId=si3.gif&_user=111111111&_pii=S0021869316300564&_rdoc=1&_issn=00218693&md5=bbe0d3ed9038a6670853ca0bf684906a" title="Click to view the MathML source">G/Pclass="mathContainer hidden">class="mathCode">$G/P$.

In this paper we study three notions of positivity for X: (1) Schubert positivity arising via Peterson's work, (2) Lusztig's total positivity and (3) Mirković–Vilonen positivity   obtained from the MV-cycles in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316300564&_mathId=si188.gif&_user=111111111&_pii=S0021869316300564&_rdoc=1&_issn=00218693&md5=e07d47ea6f2e9e7584febb43af765ae4" title="Click to view the MathML source">Gr_Gclass="mathContainer hidden">class="mathCode">${\mathrm{Gr}}_G$. The first main theorem establishes that these three notions of positivity coincide. Our second main theorem proves a parametrization of the totally nonnegative part of X, confirming a conjecture of the second author.

In type A the parametrization and relationship with Schubert positivity were proved earlier by the second author. Here we tackle the general type case and also introduce a crucial new connection with the affine Grassmannian and geometric Satake correspondence.