The (1-E)-transform in combinatorial Hopf algebras

TL;DR

This paper introduces the (1-E)-transform in combinatorial Hopf algebras, extending symmetric function endomorphisms to various algebraic structures, and explores their algebraic and representation-theoretic properties.

Contribution

It generalizes the (1-E)-transform to multiple combinatorial Hopf algebras and constructs new subalgebras with applications to representation theory and combinatorial models.

Findings

Recovered Schocker's idempotents for derangement numbers

Constructed new subalgebras of descent and Solomon-Tits algebras

Connected the transformation to solutions of the Tsetlin library problem

Abstract

We extend to several combinatorial Hopf algebras the endomorphism of symmetric functions sending the first power-sum to zero and leaving the other ones invariant. As a transformation of alphabets, this is the (1-E)-transform, where E is the exponential alphabet, whose elementary symmetric functions are e_n=1/n!. In the case of noncommutative symmetric functions, we recover Schocker's idempotents for derangement numbers [Discr. Math. 269 (2003), 239]. From these idempotents, we construct subalgebras of the descent algebras analogous to the peak algebras and study their representation theory. The case of WQSym leads to similar subalgebras of the Solomon-Tits algebras. In FQSym, the study of the transformation boils down to a simple solution of the Tsetlin library in the uniform case.