Folding Rotationally Symmetric Tableaux via Webs

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 \(\mathcal {W}\) is a web with a reflection symmetry, the corresponding tableau \(T_\mathcal {W}\) has a rotational symmetry. Folding \(T_\mathcal {W}\) transforms it into a domino tableau \(D_\mathcal {W}\). 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 \(D_\mathcal {W}\) corresponds to “\(\mathcal {W}\) modulo symmetry”.