The left-to-right minima basis of the group algebra of the symmetric group (updated version)

Abstract

We introduce a new basis of the group algebra of the symmetric group, built using the left-to-right minima sets of permutations. We show that on this basis, the descent algebra acts by triangular operators, thus making it an analogue of a cellular basis. The proof involves Dynkin elements (nested commutators) of the free algebra and their interactions with the $\mathbf{B}$-basis.

21 pages. Something between an extended abstract and a full-length paper. v2 gives a more conceptual proof of the basis property and elaborates on the eigenvalues (Section 6). Comments are welcome!