The left-to-right minima basis of the group algebra of the symmetric group (updated version)
Darij Grinberg, Ekaterina A. Vassilieva
TL;DR
The paper addresses constructing a basis for the group algebra $ A=oldsymbol{k}[S_n]$ that interacts triangulary with the Solomon descent algebra $oldsymbol{ Sigma}_n$. It introduces a poset-indexed filtration of $ A$ by right ideals $oldsymbol{R}_eta$, derived from the $oldsymbol{B}$-basis, and pairs it with a new left-to-right minima basis: for each $w\, S_n$, the element $oldsymbol{B}_{ ext{LRM}'(w)} w$ forms a basis of $ A$ with unitriangular expansion in the standard basis. The key step is establishing $oldsymbol{S}_eta = oldsymbol{R}_eta$, showing that left multiplication by any descent-algebra element acts triangularly on the LRMs-basis, with eigenvalues computable from the filtration data. Central to the proof are Dynkin elements from the free algebra and their interaction with the $oldsymbol{B}$-basis, which provide the necessary combinatorial and Hopf-algebraic framework. The results yield a concrete, basis-theoretic realization of a cellular-like structure for $ A$ compatible with $oldsymbol{ Sigma}_n$, and they pave the way for generalizations to other Coxeter groups.
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.