论文标题
偏向晶体和仙人掌群的双重性
Skew Howe duality for crystals and the cactus group
论文作者
论文摘要
有限维的复杂还原的晶体lie代数$ \ mathfrak {g} $编码其表示的结构，但也可以揭示出他们自己的新结构。我们研究了使用$ \ mathfrak {g} $的Dynkin图构建的仙人掌组$ c _ {\ Mathfrak {g}} $，及其对任何$ \ Mathfrak {g} $ - 通过Schützenberger的Crystal的组合作用。我们将此动作与贝伦斯坦 - 基里洛夫集团的动作进行了比较，涉及gelfand-tsetlin模式。 Henriques和Kamnitzer定义了$ C_N = C _ {\ Mathfrak {gl} _n} $的动作，$ n $ -tensor产品的$ \ Mathfrak {G} $ Crystals，对于任何$ \ Mathfrak {g} $ crystals of nothe上面。 We discuss the crystal corresponding to the $\mathfrak{gl}_n \times \mathfrak{gl}_m$-representation $Λ^N(\mathbb{C}^n \otimes \mathbb{C}^m),$ derive skew Howe duality on the crystal level and show that the two types of cactus group actions agree in this setting.在研究普遍包围代数的两个最大交换亚代数的最大换向子代数，参数的转移和高丁代数的两个家族中，讨论了该结果的未来应用，在该代数的转移和Gaudin代数，在该代数中，代数构建的单构作用与仙人掌群相匹配。
The crystals for a finite-dimensional complex reductive Lie algebra $\mathfrak{g}$ encode the structure of its representations, yet can also reveal surprising new structure of their own. We study the cactus group $C_{\mathfrak{g}}$, constructed using the Dynkin diagram of $\mathfrak{g}$, and its combinatorial action on any $\mathfrak{g}$-crystal via Schützenberger involutions. We compare this action with that of the Berenstein-Kirillov group on Gelfand-Tsetlin patterns. Henriques and Kamnitzer define an action of $C_n=C_{\mathfrak{gl}_n}$ on $n$-tensor products of $\mathfrak{g}$-crystals, for any $\mathfrak{g}$ as above. We discuss the crystal corresponding to the $\mathfrak{gl}_n \times \mathfrak{gl}_m$-representation $Λ^N(\mathbb{C}^n \otimes \mathbb{C}^m),$ derive skew Howe duality on the crystal level and show that the two types of cactus group actions agree in this setting. A future application of this result is discussed in studying two families of maximal commutative subalgebras of the universal enveloping algebra, the shift of argument and Gaudin algebras, where an algebraically constructed monodromy action matches that of the cactus group.
