Convolution actually can be pretty elegantly translated into multiplication by a matrix with symmetries. It's been a few years, but that was an example we had

Convolution actually can be pretty elegantly translated into multiplication by a matrix with symmetries. It's been a few years, but that was an example we had to work out by hand in my Deep Learning course at university. It's pretty much the quintessential example of an enforced symmetry, in that it introduces a symmetry against translation.