Lattices in $\mathbb F_q[[T]]^d$ and spiral shifting operators

We investigate the algebra and combinatorics of an analogue of the Hermite normal form that classifies finite-index submodules of $\mathbb F_q[[T]]^d$. We identity both normal forms as instances of Gr\"obner basis theory under different monomial orders, where the Hermite normal form corresponds to the lex order, and the new normal form the hlex order. We note that the hlex normal form recovers the Smith normal form, a feature not enjoyed by the Hermite normal form. We also identify the combinatorial structure underlying the cell decomposition induced by the hlex normal form, which appears to be of independent interest. Notably, the statistics tracking the cell dimensions is compatible, in a certain way, with a collection of $d$ ``spiral shifting operators'' on $\mathbb N^d$, which pairwise commute and collectively act freely and transitively. Using these operators, we give direct proofs of some new combinatorial identities obtained by translating the results of Solomon and Petrogradsky in terms of the hlex normal form.