UROP Proceedings 2022-23

School of Science Department of Mathematics 46 Cluster Algebra Supervisor: IP, Ivan Chi Ho / MATH Student: CHAU, Yu Foon Darin / MATH-IRE Course: UROP2100, Fall In this report, we provide some more details to the companion report submitted to SCIE3500 and generalize our result to the case with frozen variables. We also outline the proof of sign-coherence of c-vectors and gvectors. Note: We shall work with ℂ as the base field for most of the algebraic geometric constructions in this report, keeping in mind it can probably be replaced with any algebraically closed field. Cluster Algebra Supervisor: IP, Ivan Chi Ho / MATH Student: CHIU, Yan Ho / MATH-IRE Course: UROP2100, Fall UROP3100, Summer This report focuses on studying the parabolic positive representations of ( +1) using the dual graph. For any parabolic positive representations of ( +1), they can be embedded into a specific quantum torus algebra, and we can construct the dual graph from the planar quiver associated to the quantum torus algebra. Consequently, the generators ei ,fi , as well as the non-simple generators e1,n,f1,n of ( +1), can be expressed through the weighted path count of the dual graph. This paper suggests using the dual graph to understand the cluster realization of the evaluation module of the affine quantum group � � +1� in a specific parabolic group. Moreover, the dual graph allows us to obtain the cluster variables for the extra vertices and determine the assignment of arrows. Cluster Algebra Supervisor: IP, Ivan Chi Ho / MATH Student: CHOY, Sin Hang Sonia / MATH-IRE Course: UROP4100, Fall We study parabolic positive representations of the split real quantum group Uq( ℝ). In order to use geometric methods using the language of Fock-Goncharov to reprove results in [4] so as to give them geometric meaning, we wish to give apply Lusztig’s braid group action corresponding to the longest word to flip the quiver, in order to extend our maps originally on Ei and Ki to Fi and Ki’. The parabolic positive representations then correspond to the partial configuration spaces introduced in [3]. In order to do so, we establish mutation sequences for flipping certain parabolic quivers in the cases An and Bn, and give new insights into how to resolve the general case, most possibly by establishing the explicit braid group action as a sequence of mutations and permutation of vertices.