Stable Invariants of Words from Random Matrices

Let $w$ be a word in a free group. A few years ago, Magee and the first named author discovered that the stable commutator length (scl) of $w$, a well-known topological invariant, can also be defined in terms of certain Fourier coefficients of $w$-random unitary matrices [arXiv:1802.04862]. But the random-matrix side of this equality can be naturally tweaked by considering $w$-random permutations, $w$-random orthogonal matrices and so on, to produce new invariants for any given word. Are these invariants new? interesting? Do they admit an intrinsic topological description as in the case of $w$-random unitaries and scl? The current paper formalizes the definition of these invariants coming from $w$-random matrices, answers the above questions in certain cases involving generalized symmetric groups, and poses detailed conjectures in many others. In particular, we present a plethora of topological, combinatorial and algebraic invariants of words which play, or are at least conjectured to play, a similar role to the one played by scl in the above-mentioned result. Among others, these invariants include two invariants recently defined by Wilton [arXiv:2210.09853]: the stable primitivity rank and a non-oriented analog of scl.