Generators for Tensor Product Components

Let $p$ be a prime number, $F$ a field of characteristic $p$, and $G$ a cyclic group of order $q =p^a$ for some positive integer $a$. Under these circumstances every indecomposable $F G$-module is cyclic. For indecomposable $F G$-modules $U$ and $W$, we present a new recursive method for identifying a generator for each of the indecomposable components of $U \otimes W$ in terms of a particular $F$-basis of $U \otimes W$.

💡 Research Summary

The paper studies modules over a cyclic group G of order q = pᵃ where the base field F has characteristic p. In this setting every indecomposable FG‑module is cyclic and can be written as V_i (1 ≤ i ≤ q) with a Jordan block action of the generator g. For two indecomposable modules V_r and V_s the tensor product V_r⊗V_s decomposes as a direct sum of min(r,s) indecomposables V_{λ₁}⊕…⊕V_{λ_{min(r,s)}}. While the dimensions λ_ℓ are known from earlier works, explicit generators for each summand have not been given.

The author introduces a convenient basis B = {v_{i,j}=v_i⊗g^{n‑i}v_j} of V_r⊗V_s and defines subspaces D_d spanned by those basis elements with i+j=d+1. The operator (g−1) maps D_d onto D_{d‑1} in a simple way, which makes it possible to locate a generator y_ℓ inside D_{r+s‑ℓ}. The key tool is an r×s matrix J(r,s) whose anti‑diagonal entries encode the coefficients needed to express (g−1)^{λ_ℓ‑1}(y_ℓ) as a prescribed element x_{r+s+1‑ℓ‑λ_ℓ}. Lemma 1 guarantees a unique element y_{ℓ₀} for the first occurrence ℓ₀ of a given λ, and Lemma 2 shows how to obtain y_ℓ for larger ℓ by alternating signs on the same anti‑diagonal entries.

The dimensions λ_ℓ are obtained by re‑interpreting Renaud’s theorem in p‑adic language. Writing r=r_n pⁿ+R_{n‑1} and s=s_n pⁿ+S_{n‑1} with 0≤r_n,s_n<p, the paper defines m_{n‑1}=min(R_{n‑1},S_{n‑1}) and M_{n‑1}=max(R_{n‑1},S_{n‑1}). A pⁿ‑tuple W is built from the decomposition of V_{R_{n‑1}}⊗V_{S_{n‑1}}; the entries of W (denoted w_k) determine λ_ℓ via the formula λ_ℓ=(r_n+s_n−2t)pⁿ+w_k when ℓ=tpⁿ+k (1≤k≤pⁿ). This explicit description separates the cases where recursion is required (m_{n‑1}>0) from the trivial base case (m_{n‑1}=0).

When recursion is needed, the algorithm proceeds in three stages. First, it constructs an intermediate element z_ℓ in D_{r+s‑ℓ‑c pⁿ} such that (g−1)^{C‑1}(z_ℓ)=x_{r+s+1‑ℓ‑λ_ℓ}, where C and c are derived from the p‑adic data and the w_k’s. The construction of z_ℓ uses the known decompositions of the four tensor products V_{R_{n‑1}}⊗V_{S_{n‑1}}, V_{pⁿ‑R_{n‑1}}⊗V_{pⁿ‑S_{n‑1}}, V_{pⁿ‑R_{n‑1}}⊗V_{S_{n‑1}} and V_{R_{n‑1}}⊗V_{pⁿ‑S_{n‑1}} (Lemma 8). Second, the element y_ℓ is lifted from z_ℓ by solving (g−1)^{c pⁿ}(y_ℓ)=z_ℓ. This step requires inverting a k×k matrix M whose (i,j) entry is the binomial coefficient C(a+b+j−i, b+j−i). The paper applies a recent result of Nordenstam and Young, which gives an explicit formula for the inverse of such binomial matrices. Proposition 1 shows that the determinant det M is a rational integer d_k and that the adjugate matrix has integer entries; consequently M⁻¹ can be written as (det M)⁻¹·adj(M) with all entries expressed as simple rational numbers. Lemma 6 guarantees invertibility under the condition a+b<p, which is satisfied in the recursive steps.

The base case ℓ≤r+s−pⁿ+1 (so λ_ℓ=pⁿ+1) is handled directly: y_ℓ is taken as (‑1)^{r‑ℓ} v_{r+1‑ℓ, s}, and a short computation verifies that (g−1)^{pⁿ+1‑1}(y_ℓ)=x_{r+s+1‑ℓ‑pⁿ+1}. For the remaining ℓ (max(r+s−pⁿ+1,0)+1 ≤ ℓ ≤ min(r,s)) the recursive algorithm described above produces the desired generators.

Compared with earlier works, the contribution is threefold: (1) an explicit p‑adic formula for the dimensions λ_ℓ, (2) a clear description of which anti‑diagonals of J(r,s) give the generators and how to adjust signs, and (3) a concrete, non‑iterative method for lifting z_ℓ to y_ℓ using the closed‑form inverse of binomial matrices. This yields generators that are far more explicit than those obtained by Norman’s matrix recursion or by Iima‑Iwamatsu’s polynomial‑ring approach. The paper also discusses the computational complexity, noting that recursion is only needed when m_{n‑1}>0, which dramatically reduces the amount of work for many parameter choices.

In summary, the article provides a complete, constructive algorithm for producing a generator of each indecomposable component in the tensor product of two cyclic‑group modules over a field of characteristic p. The method combines p‑adic number theory, careful analysis of the (g−1) action, and modern results on binomial matrix inverses, offering both theoretical insight and practical tools for explicit module calculations.