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The crystals for a finite-dimensional complex reductive Lie algebra$\mathfrak{g}$ encode the structure of its representations, yet can also revealsurprising new structure of their own. We study the cactus group$C_{\mathfrak{g}}$, constructed using the Dynkin diagram of $\mathfrak{g}$, andits combinatorial action on any $\mathfrak{g}$-crystal via Sch\"{u}tzenbergerinvolutions. We compare this action with that of the Berenstein-Kirillov groupon 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 $\Lambda^N(\mathbb{C}^n\otimes \mathbb{C}^m),$ derive skew Howe duality on the crystal level and showthat the two types of cactus group actions agree in this setting. A futureapplication of this result is discussed in studying two families of maximalcommutative subalgebras of the universal enveloping algebra, the shift ofargument and Gaudin algebras, where an algebraically constructed monodromyaction matches that of the cactus group.

Skew Howe duality for crystals and the cactus group