Quotient groups obtained by quotienting $G^n$ by $G^{n-1}$

Question

Notation: For a group $G$, we write $G^n$ to denote the $n$-fold direct product of $G$ with itself.

Problem set up:

Consider, for a finite group $G$, and $n > 1$, the set $Q(G)_n$ of all isomorphism classes of groups of the form $\frac{G^{n}}{H_{n-1}}$, for any normal subgroup $H_{n-1}$ of $G^n$ isomorphic to $G^{n-1}$.

Note that $Q(G)_1 \subset Q(G)_2 \subset \cdots$, and so for finite groups $G$, the sequence stabilises at some $N$ - i.e. we have $Q(G)_n =Q(G)_{n+1}$ for all $n \geq N$. We call the sequence $Q(G)_1, \cdots, Q(G)_N$ the quotient series of $G$.

We recall that by the classification theorem for finite abelian groups, we can write any finite abelian group as $\bigoplus_{i = 0}^M \mathbb Z_{p_{i}^{k_i}}$ for primes $p_i$ and positive integers $k_i$.

Question: Let $G = \bigoplus_{i = 0}^M \mathbb Z_{p_{i}^{k_i}}$ be a finite abelian group. What is the quotient series of $G$?

Answer 1

I'll abbreviate $\mathbb{Z}/p^k \mathbb{Z}$ to $L_k$. Let $G = \bigoplus_{i=1}^M L_i^{a_i}$. I claim that $\bigoplus_{i=1}^N L_i^{b_i}$ is in the quotient series of $G$ if and only if the Littlewood-Richardson coefficient $$c_{(1^a_1),\ \ (2^{a_2}),\ \ \dotsc,\ \ (M^{a_M})}^{(1^{b_1} 2^{b_2} \cdots N^{b_N})}$$ is nonzero. Note the parentheses and commas: We have $M$ distinct partitions on the bottom and only one partition on the top.

You can decide whether or not this is an explicit enough answer to be useful to you.

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Proof. In general, let $\alpha$, $\beta$ and $\gamma$ be three partitions. Then there is a short exact sequence $$0 \to \bigoplus L_{\alpha_i} \to \bigoplus L_{\beta_i} \to \bigoplus L_{\gamma_i} \to 0$$ if and only if $c_{\alpha \gamma}^{\beta}$ is nonzero. This is the same as asking that there be a semistandard Young tableaux of shape $\beta/\alpha$ whose rectification is a given tableaux of shape $\gamma$. See, for example, Fulton's survey Eigenvalues, invariant factors, highest weights, and Schubert calculus. The short exact sequences of abelian groups are in Section 2; the tableaux are briefly mentioned in Section 9 and can be found in more detail in many places, such as Stanley's Enumerative Combinatorics II or Fulton's Young Tableaux.

In your situation, you want to know when we have $c_{T \alpha,\ \gamma}^{(T+1) \alpha}$ nonzero for $T$ large. So we are looking for tableaux of shape $(T+1)\alpha/T \alpha$ whose rectification is a given tableaux of shape $\gamma$. But, once $T$ is large enough, $(T+1)\alpha/T \alpha$ is just a union of disconnected rectangles. Specifically, if $\alpha = 1^{a_1} 2^{a_2} \cdots M^{a_M}$, then the connected components of $(T+1)\alpha/T \alpha$ are rectangles of shape $k \times a_k$, for $1 \leq k \leq M$. Standard properties of Young tableaux then give the answer above.