# Oscillator versus prefundamental representations II. Arbitrary higher ranks

## TL;DR

This paper explicitly determines the $oldsymbol{ ext{l}}$-weights and vectors for $q$-oscillator representations of the positive Borel subalgebra in quantum loop algebra of arbitrary rank, linking them to prefundamental representations.

## Contribution

It provides the first explicit description of $oldsymbol{ ext{l}}$-weights and vectors for these representations in higher ranks, establishing their tensor relationship with prefundamental representations.

## Key findings

- Explicit $oldsymbol{ ext{l}}$-weights and vectors for $q$-oscillator representations.
- Prefundamental representations can be constructed via tensor products of $q$-oscillator representations.
- Established the relationship between $q$-oscillator and prefundamental representations.

## Abstract

We find the $\ell$-weights and the $\ell$-weight vectors for the highest $\ell$-weight $q$-oscillator representations of the positive Borel subalgebra of the quantum loop algebra $U_q(\mathcal L(\mathfrak{sl}_{l+1}))$ for arbitrary values of $l$. Having this, we establish the explicit relationship between the $q$-oscillator and prefundamental representations. Our consideration allows us to conclude that the prefundamental representations can be obtained by tensoring $q$-oscillator representations.