Gems of combinatorics

8. Equivariant Log-Concavity of Stirling Numbers of the First Kind

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(Shiyue Li) The sequence of Stirling numbers of the first kind count the number of permutations in $S_n$ decomposing into $k$ disjoint cycles. Precisely, $$ \begin{bmatrix}n \\ k \end{bmatrix} := \#\{\sigma \in S_n \mid \sigma \text{ has } k \text{ distinct cycles}\}.$$ We say a sequence $\{a_i\}$ of nonnegative numbers $a_i$ is log-concave if $$a_i^2 \geq a_{i-1}a_{i+1} \ \forall \ i.$$ The sequence $$\left\{\begin{bmatrix}n\\k\end{bmatrix}\right\}_{k = 0}^{k=n}$$ is known to be log-concave via an analytic proof of real-rootedness of the generating polynomial.

A graded $G$-representation $$ V^\bullet = \oplus_{i = 0} V_k$$ is said to be equivariantly log-concave if, for all $i$, there exists an injection $$\varphi: V_{i-1}\otimes V_{i+1} \into V_{i} \otimes V_i $$ such that $g\cdot \varphi(v) = \varphi(g \cdot v) \all g \in G, \all v \in V_{i-1}\otimes V_{i+1}$.

Question: Is there an equivariantly log-concave $S_n$-representation $V^\bullet = \oplus V_k$ such that $|V_k| = \begin{bmatrix}n\\k\end{bmatrix}$?