On the action of Bender-Knuth generators of cactus group on the set of short semi-standard Young tableaux

Abstract

In the article by Michael Chmutov, Max Glick and Pavel Pylyavskii \cite{Chmutov} the action of the cactus group $C_N$ on the set of semi-standard Young tableaux filled with the numbers from $1$ to $N$ was defined. Namely, they constructed the set of generators (we rightfully call them Bender-Knuth generators) of the cactus group and a group homomorphism from $C_N$ to Berenstein-Kirillov group $BK_N$ (cf. \cite{Berenstein_Kirillov}), which sends these generators to the Bender-Knuth involutions on the set of semi-standard Young tableaux. In \cite{Henriques_Kamnitzer} Andre Henriques and Joel Kamnitzer defined a natural action of cactus group $C_N$ on the tensor product of $N$ normal crystals via commutors. By applying their result I defined the action of cactus group $C_N$ on the set of short semi-standard Young tableaux filled with the numbers $1, 2, \ldots, N$ in \cite{Svyatnyy}. A semi-standard Young tableau is called \textit{short} if the number of cells in the first two columns with the numbers $\leqslant N$ is less than or equal to $N$. The set of short semi-standard Young tableaux obviously forms a subset inside the set of semi-standard Young tableaux. The purpose of this paper is to explicitly compute the action of Bender-Knuth generators of cactus group $C_N$ on the set of short semi-standard Young tableaux defined in \cite{Svyatnyy} and compare it with their action on the set of semi-standard Young tableaux defined in \cite{Chmutov}.