#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### Abstract

The main goal of this dissertation is to show that the (multi-homogeneous) coordinate ring of a partial flag variety C[G/P_K^−] contains a cluster algebra for every semisimple complex algebraic group G. We use derivation properties and a canonical lifting map to prove that the cluster algebra structure A of the coordinate ring C[N_K] of a Schubert cell constructed by Goodearl and Yakimov can be lifted, in an explicit way, to a cluster structure \hat{A} living in the coordinate ring of the corresponding partial flag variety. Then we use a minimality condition to prove that the cluster algebra \hat{A} is equal to C[G/P_K^−] after localizing some special minors, which are frozen variables.

#### Date

10-25-2022

#### Recommended Citation

Kadhem, Fayadh, "A Cluster Structure on the Coordinate Ring of Partial Flag Varieties" (2022). *LSU Doctoral Dissertations*. 5975.

https://digitalcommons.lsu.edu/gradschool_dissertations/5975

#### Committee Chair

Oxley, James