论文标题
$ GL_N（\ Mathbb {C}）$的表示形式的理想倍感
Ideal Polytopes for Representations of $GL_n(\mathbb{C})$
论文作者
论文摘要
在本文中，我们将supervitential用于标志品种$ gl_n/b $和特定的坐标系统，我们称之为$ \ mathbf {i} $的理想坐标来构造polytopes $ \ m m i \ m马理{p}^{\ mathbf {\ mathbf {i}}}} $v_λ$ of $ gl_n（\ mathbb {c}）$。这里$ \ mathbf {i} $是Weyl组最长元素的表达式，$ r _+$是$ gl_n $的积极根集。 $ \ MATHCAL {P}^{\ MATHBF {i}}_λ$的晶格点可用于编码表示$v_λ$的基础。特别是，对于$ \ mathbf {i} $的特定选择，polytope $ \ mathcal {p}^{\ mathbf {i}}_λ$与gelfand-tsetlin polytope相当。多面体的构建涉及超电势的热带化。 使用Judd的作品（Arxiv：1606.06883），我们有Puiseux系列领域的超电势$ \ Mathcal {w} _ {t^λ} $的独特积极关键点。就$ \ mathbf {i} $的理想坐标而言，它的坐标在具有积极的指导术语的意义上是积极的。我们的新polytopes的出色属性与此关键点的热带版本有关，对于$ \ m \ m \ mathbf {i} $的每一个选择，它都在$ \ mathbb {r}^{r}^{r _+} $中给出了一个点，它在polytope $ \ \ m m mathcal {p} p}^^$} $}^polytope $ \ mathcal {我们证明，这个热带临界点独立于减少的表达式$ \ mathbf {i} $，并且由Judd引入的$λ$的理想填充模式给出。 最后，将这些结果与Rietsch（Arxiv：Math/0511124）的工作相结合，将超电势的临界点与Toeplitz矩阵相关联，我们表明，对于完全阳性的下三角toeplitz矩阵，Puiseux系列的PUISEUX系列而在PUISE串联系列中分解为简单的根系，以使因子填充了一个理想的填充物。
In this paper we use the superpotential for the flag variety $GL_n/B$ and particular coordinate systems that we call ideal coordinates for $\mathbf{i}$, to construct polytopes $\mathcal{P}^{\mathbf{i}}_λ$ inside $\mathbb{R}^{R_+}$, associated to highest weight representations $V_λ$ of $GL_n(\mathbb{C})$. Here $\mathbf{i}$ is a reduced expression of the longest element of the Weyl group and $R_+$ is the set of positive roots of $GL_n$. The lattice points of $\mathcal{P}^{\mathbf{i}}_λ$ can be used to encode a basis of the representation $V_λ$. In particular, for a specific choice of $\mathbf{i}$, the polytope $\mathcal{P}^{\mathbf{i}}_λ$ is unimodularly equivalent to a Gelfand-Tsetlin polytope. The construction of the polytopes involves tropicalisation of the superpotential. Using work of Judd (arXiv:1606.06883) we have that there is a unique positive critical point of the superpotential $\mathcal{W}_{t^λ}$ over the field of Puiseux series. Its coordinates, in terms of the ideal coordinates for $\mathbf{i}$, are positive in the sense of having positive leading term. The remarkable property of our new polytopes relates to the tropical version of this critical point, which, for every choice of $\mathbf{i}$, gives a point in $\mathbb{R}^{R_+}$ that lies in the interior of the polytope $\mathcal{P}^{\mathbf{i}}_λ$. We prove that this tropical critical point is independent of the reduced expression $\mathbf{i}$, and that it is given by a pattern called the ideal filling for $λ$ that was introduced by Judd. Finally, combining these results with work of Rietsch (arXiv:math/0511124) relating critical points of the superpotential with Toeplitz matrices, we show that for a totally positive lower-triangular Toeplitz matrix over the field of Puiseux series factorized into simple root subgroups, the valuations of the factors give an ideal filling.
