Explicit description of the degree function in terms of quantum Lakshmibai-Seshadri paths

TL;DR

This paper provides an explicit, computable description of the degree function for quantum Lakshmibai-Seshadri paths using the parabolic quantum Bruhat graph, impacting the understanding of energy functions in crystal bases.

Contribution

It introduces a new explicit formula for the degree function in terms of the parabolic quantum Bruhat graph, applicable to quantum Lakshmibai-Seshadri paths and related structures.

Findings

Explicit description of the degree function in terms of the quantum Bruhat graph

Computable formula for the global energy function on tensor products of crystals

Application to classically restricted one-dimensional sums

Abstract

We give an explicit and computable description, in terms of the parabolic quantum Bruhat graph, of the degree function defined for quantum Lakshmibai-Seshadri paths, or equivalently, for "projected" (affine) level-zero Lakshmibai-Seshadri paths. This, in turn, gives an explicit and computable description of the global energy function on tensor products of Kirillov-Reshetikhin crystals of one-column type, and also of (classically restricted) one-dimensional sums.