Relationship between Theorem 3.20 and the Steinberg tensor product formula for nonzero carry patterns

Determine the precise relationship between the factorization Ic,d = ∏r (m^{|δ_{t_r}|})^{[p^{t_r}]} provided by Theorem 3.20 for carry ideals Ic,d in S = k[x, y] (char k = p > 0) and the Steinberg tensor product formula L(a) ≅ ⊗_{r=0}^M L(a_r)^{[p^r]} for GL2(k)-representations, in the case where the carry pattern c ∈ C(d, 2, p) is not (0, …, 0).

Background

The paper classifies GLn(k)-invariant ideals in characteristic p via carry patterns and, in two variables, gives an explicit description (Theorem 3.20) expressing a carry ideal Ic,d as a product of Frobenius powers of powers of the homogeneous maximal ideal. This decomposition depends on the type c segmentation of the degree d.

Remark 3.22 compares this algebraic factorization with the Steinberg tensor product formula for GL2(k)-representations. When c = (0, …, 0), Ic,d is generated by L(d) ⊂ Sd, and in certain cases the product decomposition of Ic,d aligns with the Steinberg factorization. However, for general nonzero carry patterns c, the authors explicitly note that the relationship between Theorem 3.20’s decomposition and the Steinberg formula is not understood.