Multinomial fix-Mahonian statistics

The permutation statistics fix, des, maj, and inv have different original contexts, and appear in diverse scientific domains such as probability, physics, and genomics. But so far, they only meet together in generating functions and equidistributions. Examples are the generating function of (inv, des, maj) computed by Garsia and Gessel, and the equidistributivity of (fix, des, maj) and (fix, dez, maz) proved by Foata and Han. Recently, Tsilevich and Vershik determined the eigenvalues and multiplicities of (des(σ τ-1))σ, τ ∈ 𝔖n, (maj(σ τ-1))σ, τ ∈ 𝔖n, and (inv(σ τ-1))σ, τ ∈ 𝔖n, and Tsilevich determined those of (fix(σ τ-1))σ, τ ∈ 𝔖n. This article studies combinations of these statistics in terms of matrices. For that, the regular representation of the sum over all permutations weighted by the sum of their multinomial descents, inversions, and fixed points is considered. We compute the eigenvalues and multiplicities of that matrix. Then, we deduce those of (des(σ τ-1) + maj(σ τ-1) + inv(σ τ-1) + fix(σ τ-1))σ, τ ∈ 𝔖n.