We construct a complete system of primitive orthogonal idempotents and give an explicit quiver presentation of the monoid algebra of the stylic monoid introduced by Abram and Reutenauer [AR22].

Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.

We give a direct proof of Schensted's lemma asserting that row and column insertion in a tableau commute.

Contents 1. Introduction 1 2. Ordering circular permutations 3 3. An isomorphism towards admitted vectors 7 4. Properties of the poset 18 4.1. Lattice 18 4.2. Multiplicities in the Hasse diagram and Eulerian numbers 19 4.3. Limiting poset 21 5. The functions δ and triangulations of an n-gon 22 6. An interval in the affine symmetric group 23 6.1. Preliminaries on Sn 23 6.2. A poset isomorphism 24 7. lines diagrams 30 References 33