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A randomisation of the Berele insertion algorithm is proposed, where theinsertion of a letter to a symplectic Young tableau leads to a distributionover the set of symplectic Young tableaux. Berele's algorithm provides abijection between words from an alphabet and a symplectic Young tableau alongwith a recording oscillating tableau. The randomised version of the algorithmis achieved by introducing a parameter $0 < q < 1$. The classic Berelealgorithm corresponds to letting the parameter $q \to 0$. The new versionprovides a probabilistic framework that allows to prove Littlewood-typeidentities for a $q$-deformation of the symplectic Schur functions. Thesefunctions correspond to multilevel extensions of the continuous $q$-Hermitepolynomials. Finally, we show that when both the original and the $q$-modifiedinsertion algorithms are applied to a random word then the shape of thesymplectic Young tableau evolves as a Markov chain on the set of partitions.

A $q$-deformation of the symplectic Schur functions and the Berele insertion algorithm