Directed Laplacians of R-trivial Monoids: Integer Spectra, Eulerian Multiplicities, and the Palindromic Dichotomy

Let M be a finite R-trivial aperiodic monoid generated by K idempotents, and let L(M) denote the outLaplacian of its right Cayley graph. We develop a spectral theory for L(M) complementary to, andindependent of, the stochastic theory of Ayyer–Schilling–Steinberg–Thiéry.Our first result is that L(M) is upper-triangular in any linear extension of the J-order on M;consequently its spectrum is the integer multiset {outdeg(x) : x in M} in {0, 1, ..., K}. This is theInteger Spectrum Theorem.Specializing to the 0-Hecke monoid H_0(A_K) of type A_K (which has K generators and |H_0(A_K)|= (K+1)!), we show that the multiplicity of eigenvalue j equals the Eulerian number A(K, j), countingpermutations in S_{K+1} with exactly j descents. As a corollary, the eigenvalues of L(H_0(A_K)) areexactly half those of the undirected Laplacian of Cay(S_{K+1}, generators), verified for K ≤ 4.Finally, we introduce the ternary relation cube: the family of 27 finitely presented monoids on threeidempotent generators where each pairwise relation is one of braid (B), absorption (A), orcommutation (C). Of the 10 orbits under S_3-permutation of generators, nine are finite. We prove thePalindromic Dichotomy: a monoid in this family has palindromic directed spectrum if and only if it isJ-trivial, if and only if its fix-rate equals 1/2, if and only if it lies on the Coxeter face {B, C}^3.Keywords:R-trivial monoid, 0-Hecke monoid, directed Cayley graph, Laplacian spectrum, Eulerian numbers, J-trivial,palindromic spectrum, descent statisticsMSC 2020:Primary 20M30; Secondary 05A05, 05E10, 15A18, 60J10