Classification of degree-n Riordan representations of S3 over Z3

Classify all degree-n Riordan representations of the symmetric group S3 over the ring of integers modulo 3 (Z3) for all integers n ≥ 3, where a degree-n Riordan representation is obtained by composing a representation by Riordan arrays over Z3 with the truncation map R(Z3) → Rn(Z3) as defined by πn in equation (3.4).

Background

The paper proves that S3 admits faithful representations by Riordan arrays over Z3 (and more generally over GF(3^q)) and completely classifies the degree-2 case, showing exactly two nonequivalent degree-2 induced faithful representations. This motivates extending the classification beyond degree 2.

The authors define truncated Riordan groups Rn(K) via the epimorphism πn: R(K) → Rn(K) and use this to induce finite-dimensional linear representations from Riordan array representations. The question asks for a full classification of such representations for S3 when the truncation degree n ≥ 3.