Some combinatorial and group theoretical studies motivated by Physics

A survey of algebra q-deformations that were encountered in the context of physical applications will be presented. We start from the non-commutative plane giving rise to the q-binomial coefficients and to the q-exponential, and consider the problem of normal ordering of the q-oscillator giving rise to the q-Stirling numbers. We show how the Hecke algebra, being a q-deformation of the symmetric group, yields useful insights into the representation theory of the latter. Similar phenomena will be presented for the q-Casimir operators of the quantum unitary groups.