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The understanding of the topology of the spectra of quantum Schubert cellalgebras hinges on the description of their prime factors by ideals invariantunder the maximal torus of the ambient Kac-Moody group. We give an explicitdescription of these prime quotients by expressing their Cauchon generators interms of sequences of normal elements in chains of subalgebras. Based on this,we construct large families of quantum clusters for all of these algebras andthe quantum Richardson varieties associated to arbitrary symmetrizableKac-Moody algebras and all pairs of Weyl group elements. Along the way wedevelop a quantum version of the Fomin-Zelevinsky twist map for all quantumRichardson varieties. Furthermore, we establish an explicit relationshipbetween the Goodearl-Letzter and Cauchon approaches to the descriptions of thespectra of symmetric CGL extensions.

Prime factors of quantum Schubert cell algebras and clusters for quantum Richardson varieties