完全非负最大环形群与对立布雷特区间
Totally nonnegative maximal tori and opposed Bruhat intervals
AI总结：
本文验证了Lusztig关于完全正部分到完全正最大环形群的映射猜想，并研究了完全非负布雷特子群与完全非正布雷特子群的对立关系，同时将完全正最大环形群与幅度hedron联系起来。
摘要：
Lusztig (2024)最近引入了代数群G的完全正最大环形群空间$\mathcal{T}_{>0}$。每个这样的环形群是完全正伯尔子群和完全负伯尔子群的交集。Lusztig定义了一个从G的完全正部分到$\mathcal{T}_{>0}$的映射，并猜测该映射是满射的。我们验证了这一猜想。我们还研究了$\mathcal{T}_{>0}$的闭包，通过研究完全非负伯尔子群与完全非正伯尔子群何时相对立。我们的主要结果将该问题归结为韦伊群$W$的布鲁hat区间对之间的新组合关系，我们称之为'对立'。我们为$G = \text{SL}_n$（且$W$是对称群）提供了对立的特征化。在此过程中，我们推翻了Lusztig (2021)关于完全非负伯尔子群的另一个猜想。最后，我们通过展示$\mathcal{T}_{>0}$可以被视为'通用旗振幅hedron'来将$\mathcal{T}_{>0}$与Arkani-Hamed和Trnka (2014)在理论物理中引入的振幅hedron联系起来。这进一步激励了对$\mathcal{T}_{>0}$及其闭包的研究。
英文摘要：
Lusztig (2024) recently introduced the space $\mathcal{T}_{>0}$ of totally positive maximal tori of an algebraic group $G$. Each such torus is the intersection of a totally positive Borel subgroup and a totally negative Borel subgroup. Lusztig defined a map from the totally positive part of $G$ to $\mathcal{T}_{>0}$ and conjectured that it is surjective. We verify this conjecture. We also examine the closure of $\mathcal{T}_{>0}$, by studying when a totally nonnegative Borel subgroup is opposed to a totally nonpositive Borel subgroup. Our main result reduces this problem to a new combinatorial relation between pairs of Bruhat intervals of the Weyl group $W$, which we call 'opposition'. We provide a characterization of opposition when $G = \text{SL}_n$ (and $W$ is the symmetric group). Along the way, we disprove another conjecture of Lusztig (2021) on totally nonnegative Borel subgroups. Finally, we connect $\mathcal{T}_{>0}$ to the amplituhedron introduced by Arkani-Hamed and Trnka (2014) in theoretical physics, by showing that $\mathcal{T}_{>0}$ can be regarded as a 'universal flag amplituhedron'. This gives further motivation for studying $\mathcal{T}_{>0}$ and its closure.
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