math.QA — Quantum Algebra
We prove Conjecture 11 of Lee, Li, Rupel, and Zelevinsky on the support of triangular basis elements in rank-two quantum cluster algebras.
math.AG — Algebraic Geometry
We proved the predictions by Ben-Zvi, Sakellaridis, and Venkatesh that the Plancherel algebra is commutative and its loop-rotated version is flat over .
math.CO — Combinatorics
Burman and Kulishov defined Lie elements in the group algebra by comparing, on every exterior power of the reflection representation , the usual action of with the infinitesimal action induced by its action on . They conjectured that the Lie algebra of all Lie elements is generated by the Kirchhoff differences . We disprove this conjecture for by exhibiting an explicit counterexample arising from the -block of . More generally, we describe in terms of the Artin--Wedderburn decomposition of : its hook blocks are determined by the action on , whereas its non-hook blocks are arbitrary. Consequently, the primitive central idempotents of the non-hook blocks yield linearly independent obstructions to the conjecture. We also identify the Lie algebra generated by the Kirchhoff differences in terms of the derived algebra of the Lie algebra generated by transpositions. Along the way, we give an integral, and hence characteristic-free, proof that the exterior powers of are the hook-shaped Specht modules.
math.QA — Quantum Algebra
We prove quantum Laurent positivity for mutation-acyclic skew-symmetrizable cluster algebras of arbitrary finite rank.