# Hall-Littlewood RSK field

## TL;DR

This paper introduces a randomized Hall-Littlewood RSK algorithm and a new probabilistic model called the Hall-Littlewood RSK field, unifying and extending various known integrable models and providing new combinatorial and probabilistic insights.

## Contribution

It presents the first randomized Hall-Littlewood RSK algorithm and the Hall-Littlewood RSK field, connecting known models and establishing new combinatorial properties.

## Key findings

- Introduced the Hall-Littlewood RSK field as a new probabilistic model.
- Derived formulas for observables extending Macdonald process results.
- Established combinatorial properties like invertibility and symmetry.

## Abstract

We introduce a randomized Hall-Littlewood RSK algorithm and study its combinatorial and probabilistic properties. On the probabilistic side, a new model --- the Hall-Littlewood RSK field --- is introduced. Its various degenerations contain known objects (the stochastic six vertex model, the asymmetric simple exclusion process) as well as a variety of new ones. We provide formulas for a rich class of observables of these models, extending existing results about Macdonald processes. On the combinatorial side, we establish analogs of properties of the classical RSK algorithm: invertibility, symmetry, and a "bijectivization" of the skew-Cauchy identity.