Berenstein, ArkadyRupel, Dylan2012-10-262012-10-262012https://hdl.handle.net/1794/12400We de ne the quantum cluster character assigning an element of a quantum torus to each representation of a valued quiver (Q; d) and investigate its relationship to external and internal mutations of a quantum cluster algebra associated to (Q; d). We will see that the external mutations are related to re ection functors and internal mutations are related to tilting theory. Our main result will show the quantum cluster character gives a cluster monomial in this quantum cluster algebra whenever the representation is rigid, moreover we will see that each non-initial cluster variable can be obtained in this way from the quantum cluster character.en-USAll Rights Reserved.ClusterQuantumQuiverTiltingElectronic Thesis or Dissertation