Robinson-Schensted algorithms and quantum stochastic double product integrals
This thesis is divided into two parts.\ud \ud In the first part (Chapters 1, 2, 3) various Robinson-Schensted (RS) algorithms are discussed. An introduction to the classical RS algorithm is presented, including the symmetry property, and the result of the algorithm Doob h-transforming the kernel from the Pieri rule of Schur functions h when taking a random word [O'C03a]. This is followed by the extension to a q-weighted version that has a branching structure, which can be alternatively viewed as a randomisation of the classical algorithm. The q-weighted RS algorithm is related to the q-Whittaker functions in the same way as the classical algorithm is to the Schur functions. That is, when taking a random word, the algorithm Doob h-transforms the Hamiltonian of the quantum Toda lattice where h are the q-Whittaker functions. Moreover, it can also be applied to model the q-totally asymmetric simple exclusion process introduced in [SW98]. Furthermore, the q-RS algorithm also enjoys a symmetry property analogous to that of the symmetry property of the classical algorithm. This is proved by extending Fomin's growth diagram technique [Fom79, Fom88, Fom94, Fom95], which covers a family of the so-called branching insertion algorithms, including the row insertion proposed in [BP13].\ud \ud In the second part (Chapters 4, 5) we work with quantum stochastic analysis. First we introduce the basic elements in quantum stochastic analysis, including the quantum probability space, the momentum and position Brownian motions [CH77], and the relation between rotations and angular momenta via the second quantisation, which is generalised to a family of rotation-like operators [HP15a]. Then we discuss a family of unitary quantum causal stochastic double product integrals E, which are expected to be the second quantisation of the continuous limit W of a discrete double product of aforementioned rotation-like operators. In one special case, the operator E is related to the quantum Levy stochastic area, while in another case it is related to the quantum 2-d Bessel process. The explicit formula for the kernel of W is obtained by enumerating linear extensions of partial orderings related to a path model, and the combinatorial aspect is closely related to generalisations of the Catalan numbers and the Dyck paths. Furthermore W is shown to be unitary using integrals of the Bessel functions.