Folding rotationally symmetrical tableaux via webs

TL;DR

This paper explores the relationship between symmetric webs and folded tableaux, providing algorithms that connect webs with domino tableaux through symmetry operations, enhancing understanding of their combinatorial structures.

Contribution

It introduces explicit algorithms linking symmetric webs to domino tableaux, clarifying how folding webs corresponds to tableau transformations.

Findings

Folding symmetric webs corresponds to folding tableaux along symmetry axes.

Algorithms establish bijections between symmetric webs and domino tableaux.

The methods clarify the combinatorial relationship between web symmetries and tableau structures.

Abstract

Rectangular standard Young tableaux with 2 or 3 rows are in bijection with $U_q(\mathfrak{sl}_2)$-webs and $U_q(\mathfrak{sl}_3)$-webs respectively. When $W$ is a web with a reflection symmetry, the corresponding tableau $T_W$ has a rotational symmetry. Folding $T_W$ transforms it into a domino tableau $D_W$. We study the relationships between these correspondences. For 2-row tableaux, folding a rotationally symmetric tableau corresponds to "literally folding" the web along its axis of symmetry. For $3$-row tableaux, we give simple algorithms, which provide direct bijective maps between symmetrical webs and domino tableaux (in both directions). These details of these algorithms reflect the intuitive idea that $D_W$ corresponds to "$W$ modulo symmetry".