Abstract
Triangular fully packed loop configurations (TFPLs) emerged as auxiliary objects in the study of fully packed loop configurations on a square (FPLs) corresponding to link patterns with a large number of nested arches. Wieland gyration, on the other hand, was invented to show the rotational invariance of the numbers $A_\pi$ of FPLs corresponding to a given link pattern $\pi$. The focus of this article is the definition and study of Wieland drift on TFPLs. We show that the repeated application of this operation eventually leads to a configuration that is left invariant. We also provide a characterization of such stable configurations. Finally we apply Wieland drift to the study of TFPL configurations, in particular giving new and simple proofs of several results.
Highlights
Triangular fully packed loop configurations (TFPLs) first appeared in the study of ordinary fully packed loop configurations (FPLs)
It soon turned out that TFPLs possess a number of nice properties, which made them worthy objects of study by themselves
Packed loop configurations first came up in statistical physics; they are an alternative representation of six-vertex model configurations which are in one-to-one correspondence with square-ice configurations, see for example [1] and [9]
Summary
Triangular fully packed loop configurations (TFPLs) first appeared in the study of ordinary fully packed loop configurations (FPLs). It soon turned out that TFPLs possess a number of nice properties, which made them worthy objects of study by themselves They can be seen as a generalized model for Littlewood–Richardson coefficients, thereby establishing an unexpected link to algebra. The Wieland gyration was used to prove the rotational invariance of the numbers Aπ of FPLs corresponding to a given link pattern π It was later heavily used by Cantini and Sportiello [2] to prove the Razumov–Stroganov conjecture. It is introduced in Definition 7 and we give its first properties, culminating in Theorem 13.
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