The Action of the Symmetric Group on a Generalized Partition Semilattice

Abstract

Given an integer $n\geq 2$, and a non-negative integer $k$, consider all affine hyperplanes in ${\bf R}^n$ of the form $x_i=x_j +r$ for $i,j\in[n]$ and a non-negative integer $r\leq k$. Let $\Pi_{n,k}$ be the poset whose elements are all nonempty intersections of these affine hyperplanes, ordered by reverse inclusion. It is noted that $\Pi_{n,0}$ is isomorphic to the well-known partition lattice $\Pi_n$, and in this paper, we extend some of the results of $\Pi_n$ by Hanlon and Stanley to $\Pi_{n,k}$.

Just as there is an action of the symmetric group ${S}_n$ on $\Pi_n$, there is also an action on $\Pi_{n,k}$ which permutes the coordinates of each element. We consider the subposet $\Pi_{n,k}^\sigma$ of elements that are fixed by some $\sigma\in {S}_n$, and find its Möbius function $\mu_\sigma$, using the characteristic polynomial. This generalizes what Hanlon did in the case $k=0$. It then follows that $(-1)^{n-1}\mu_\sigma(\Pi_{n,k}^\sigma)$, as a function of $\sigma$, is the character of the action of ${S}_n$ on the homology of $\Pi_{n,k}$.

Let $\Psi_{n,k}$ be this character times the sign character. For ${C}_n$, the cyclic group generated by an $n$-cycle $\sigma$ of ${S}_n$, we take its irreducible characters and induce them up to ${S}_n$. Stanley showed that $\Psi_{n,0}$ is just the induced character $\chi\uparrow_{{C}_n}^{{S}_n}$ where $\chi(\sigma)=e^{2\pi i/n}$. We generalize this by showing that for $k>0$, there exists a non-negative integer combination of the induced characters described here that equals $\Psi_{n,k}$, and we find explicit formulas. In addition, we show another way to prove that $\Psi_{n,k}$ is a character, without using homology, by proving that the derived coefficients of certain induced characters of ${S}_n$ are non-negative integers.