Intrinsic energy is a loop Schur function

TL;DR

This paper presents a new explicit formula for the energy function in certain crystal tensor products, linking it to loop Schur functions and total positivity in loop groups.

Contribution

It introduces a subtraction-free formula for the energy function as the tropicalization of a loop Schur function, connecting crystal theory with total positivity.

Findings

Energy function expressed as tropicalization of loop Schur function

Explicit subtraction-free formula derived for symmetric powers

Links between crystal energy and total positivity established

Abstract

We give an explicit subtraction-free formula for the energy function in tensor products of Kirillov-Reshetikhin crystals for symmetric powers of the standard representation of U_q'(\hat sl_n). The energy function is shown to be the tropicalization of a stretched staircase shape loop Schur function. The latter were introduced by the authors in the study of total positivity in loop groups.