RNA, local moves on plane trees, and transpositions on tableaux

TL;DR

This paper introduces functions on plane trees inspired by RNA local moves and symmetric group transpositions, revealing their algebraic structure and biological relevance, including connections to Young tableaux and Weyl groups.

Contribution

It defines new functions modeling RNA local moves, linking them to algebraic structures like symmetric group actions and Weyl groups, and analyzes their graph properties.

Findings

The $s_i$ functions form a structure close to a group action on plane trees.

The graph of $s_i$-local moves is a connected, graded poset with unique extremal elements.

The extended $s_i^C$ functions produce a graph with two connected components, reflecting symmetry properties.

Abstract

We define a collection of functions $s_i$ on the set of plane trees (or standard Young tableaux). The functions are adapted from transpositions in the representation theory of the symmetric group and almost form a group action. They were motivated by $\textit{local moves}$ in combinatorial biology, which are maps that represent a certain unfolding and refolding of RNA strands. One main result of this study identifies a subset of local moves that we call $s_i$-local moves, and proves that $s_i$-local moves correspond to the maps $s_i$ acting on standard Young tableaux. We also prove that the graph of $s_i$-local moves is a connected, graded poset with unique minimal and maximal elements. We then extend this discussion to functions $s_i^C$ that mimic reflections in the Weyl group of type $C$. The corresponding graph is no longer connected, but we prove it has two connected components, one of symmetric and the other of asymmetric plane trees. We give open questions and possible biological interpretations.