We analyse models of Matrix Quantum Mechanics in the double scaling limit that contain non-singlet states. The finite temperature partition function of such systems contains non-trivial winding modes (vortices) and is expressed in terms of a group theoretic sum over representations. We then focus on the model of Kazakov-Kostov-Kutasov when the first winding mode is dominant. In the limit of large representations (continuous Young diagrams), and depending on the values of the parameters of the model such as the compactification radius and the string coupling, the dual geometric background corresponds either to that of a long string (winding mode) condensate or a 2d (non-supersymmetric) semi-classical Black Hole competing with the thermal linear dilaton background. In the matrix model we are free to tune these parameters and explore various regimes of this phase diagram. Our construction allows us to identify the origin of the microstates of the long string condensate/2d Black Hole arising from the non trivial representations.