The center of small quantum groups I: the principal block in type A

Abstract

We develop an elementary algebraic method to compute the center of the principal block of a small quantum group associated with a complex semisimple Lie algebra at a root of unity. The exemplary case of ${\mathfrak{sl}}_3$ is computed explicitly, and further evidence of ${\mathfrak{sl}}_4$ is sketched. This allows us to formulate the conjecture that, as a bigraded vector space, the center of a regular block of the small quantum ${\mathfrak{sl}}_m$ at a root of unity is isomorphic to Haiman's diagonal coinvariant algebra for the symmetric group $S_m$.

44 pages, V3 contains corrections from the referee and updated references, comments welcome, International Mathematics Research Notices 2017

References in corpus (3)

• The small quantum group and the Springer resolution
• The center of the small quantum group II: singular blocks
• The null-cone and cohomology of vector bundles on flag manifolds

Cited by in corpus (5)

• Remarks on the derived center of small quantum groups
• A computer algorithm for the BGG resolution
• Affine Springer Fibers, Procesi bundles, and Cherednik algebras
• The center of the small quantum group u_q(sl_3) for l=5: a numerical computation
• On certain Hochschild cohomology groups for the small quantum group