Existence of a compositional law for prime-power matrices

Construct a binary operation [?] on the matrices M(p^e), for a fixed prime p, such that the product M(p^e) [?] M(p^{e'}) is M(p^{e+e'}) and, in particular, M(p)^{[?]e}=M(p^e), thereby providing a composition law analogous to the Kronecker-product identity for coprime indices.

Background

For coprime integers k1 and k2, the paper proves that M(k1k2) is the Kronecker product M(k1) [?] M(k2). Section 5.3 asks whether an analogous construction exists when the index is a power of a single prime, where the matrices M(pe) have sizes e+1 rather than dimensions that multiply under tensor products.

The authors examine the symmetric product SeM(p) as a natural candidate for deriving M(pe) from M(p), but show that M(p2) is not conjugate to S2M(p) by comparing characteristic polynomials. This rules out that particular construction, while leaving unresolved whether some other law with the required composition property exists.

References

It is tempting to ask if there could be such results when k = pe is a power of a prime p, with the Kronecker product replaced by a law [?] such that the product M(pe) [?] M(p{e'}) of M(pe), of size e + 1, and indexed by D_{pe}, times M(p{e'}), of size e' + 1, is the matrix M(p{e+e'}) of size e + e' + 1 indexed by D_{p{e+e'}}. In particular, one would like to have M(p){[?]e} = M(pe).

— Number of partitions of modular integers (with an Appendix by P. Deligne)  (2502.19523 - Broadhurst et al., 26 Feb 2025) in Section 5.3, p. 16