On the fermionc formula and the Kirillov-Reshetikhin conjecture

TL;DR

This paper proves the Kirillov-Reshetikhin fermionic formula for certain modules of quantum affine algebras, confirming the conjecture in many cases, especially for most nodes of exceptional Lie algebras.

Contribution

It establishes the validity of the fermionic formula for modules where the corresponding simple root appears with multiplicity at most 2 in the highest root.

Findings

Proves the conjecture for modules W(mλ_i) under specified conditions.

Confirms the conjecture for all but a few nodes in exceptional algebras.

Advances understanding of tensor product decompositions in quantum affine algebra representations.

Abstract

The fermionic formula conjectured by Kirillov and Reshetikhin describes the decomposition (as a module for $U_q(\frak g)$) of a tensor product of multiples of of fundamental representations $W(m\lambda_i)$ of the corresponding quantum affine algebras. In this paper, we show that the conjecture is true for the modules W(m\lambda_i), if $i$ is such that the corresponding simple root occurs in the highest root of the simple Lie algebra with multiplicity at most 2. In particular, the conjecture is established for all but a few nodes for the exceptional algebras.