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We study the $n$-component priority asymmetric simple exclusion process($n$-ASEP) with reflecting boundaries. We obtain all invariant measures inexplicit form and prove reversibility. Using the symmetry of the generator ofthe process under the quantum algebra $U_q[\mathfrak{gl}(n+1)]$ we constructduality functions with respect to which the $n$-ASEP is self-dual, both for thefinite and the infinite integer lattice. For the $n$-ASEP on the infinitelattice we use self-duality to derive in explicit form the time evolution of afamily of measures with $K$ shocks in terms of the transition probability of$K$ coloured particles in a shock exclusion process with particle-dependenthopping rates and nearest-neighbour colour exchange. This process is a gas ofparticles that forms a bound state, corresponding to shock coalescence onmacroscopic scale.

Self-duality and shock dynamics in the $n$-component priority ASEP