Spectral Degeneracies in the Totally Asymmetric Exclusion Process

We study the spectrum of the Markov matrix of the totally asymmetric exclusion process (TASEP) on a one-dimensional periodic lattice at arbitrary filling. Although the system does not possess obvious symmetries except translation invariance, the spectrum presents many multiplets with degeneracies of high order when the size of the lattice and the number of particles obey some simple arithmetic rules. This behaviour is explained by a hidden symmetry property of the Bethe Ansatz. Assuming a one-to-one correspondence between the solutions of the Bethe equations and the eigenmodes of the Markov matrix, we derive combinatorial formulae for the orders of degeneracy and the number of multiplets. These results are confirmed by exact diagonalisations of small size systems. This unexpected structure of the TASEP spectrum suggests the existence of an underlying large invariance group.