Maximal zero-sum free sequences of $C_3^n$

Question

I am working on the Davenport constant for groups, $D(G)$, which is the minimal number $d$ such that every sequence or multiset of $d$ elements of the group $G$ always contains some non-empty zero-sum subsequence.

For abelian groups, we know a lot of result and facts. In particular, $D(C_3^n)=1+2n$ so that maximal zero-sum free sequences have length $2n$. For example, the sequence consisting of each vector $e_i$ twice. Do we know exactly the structure of all maximal zero-sum free sequences?

UPDATE (2024-09-05). As Salvo Tringali states (thank you!) in the answer to this question, Prof. Geroldinger shared the characterisation for $n=3$.

As a summary, a sequence $U$ of length 7 in $C_3^3$ is a minimal zero-sum sequence of maximal length over $C_3^3$ if and only if there exists a basis $(e_1, e_2, e_3)$ of $C_3^3$ and $a_i, b_j \in [0,2]$ for $i \in [1,5]$ and $j \in [1,3]$ with $\sum_{i=1}^{5} a_i \equiv \sum_{j=1}^{3} b_j \equiv 1 (mod 3)$ such that

$U= e_1^2 * \prod_{i=1}^{2} (a_i e_1 + e_2) * \prod_{j=1}^{3} (a_{2+j} e_1 + b_j e_2 + e_3)$.

If I am not wrong, a sequence is maximal zero-sum free sequence if and only if it is a subsequence of length 6 of some minimal zero-sum sequence of maximal length 7 (characterisation above). If this is true (otherwise, please correct me) and I have "some flexibility" of choice of elements (for example, I have 4 fixed elements and flexibility of choice between two elements for remaining two elements in $C_3^3$), can I always ensure that some combination makes it NOT maximal zero-sum free sequence? Does someone come up with some corollary about the form of maximal zero-sum free sequences? Thank you very much!!

PD: If you are curious about why am I asking this question, the idea is to prove that any sequence of 14 elements in $S_3^3$ gives some one-product subsequence. If we have 2 elements of same "type", with 8 types (odd/even, odd/even, odd/even), then the product of both elements is in $A_3^3 \cong C_3^3$, and if elements are distinct then the order of the product matters, getting distict elements. Thus, in many situations we can get 6 elements in $A_3^3$ taking products of these 14 elements, with some flexibility of choice (if we get $D(C_3^3)=7$, we are done).

Answer 1

I sent an email to Alfred Geroldinger with a link to this thread. Here is a summary of his reply (I'm posting with his permission):

• The structure of minimal zero-sum sequences of maximal length over $C_3^n$ is unknown.
• Lemma 4.2 in [Geroldinger and Schmid, J. Korean Math. Soc. 56 (2019), No. 4, 869-915. DOI: 10.4134/JKMS.j180467] gives the structure of minimal zero-sum sequences of maximal length over Abelian groups of rank two.

UPDATE (2024-08-29). As I've just learned from Alfred Geroldinger, the structure of minimal zero-sum sequences of maximal length over $C_3^3$ is the subject of Lemma 5.4 in [Geroldinger, Grynkiewicz, and Schmid, J. Théorie Nombres Bordeaux 23 (2011), Nr. 1, 137-169. EuDML: https://eudml.org/doc/219687]. This answers a question by Mikel Martinez Puente in the comments on this post.