Rook sums in the symmetric group algebra

Abstract

Let $\mathcal{A}$ be the group algebra $\mathbf{k}[S_n]$ of the $n$-th symmetric group $S_n$ over a commutative ring $\mathbf{k}$. For any two subsets $A$ and $B$ of $[n]$, we define the elements \[ \nabla_{B,A}:=\sum_{\substack{w\in S_n;\\w\left( A\right) =B}} w \qquad \text{and} \qquad \widetilde{\nabla}_{B,A}:=\sum_{\substack{w\in S_n;\\w\left( A\right) \subseteq B}}w \] of $\mathcal{A}$. We study these elements, showing in particular that their minimal polynomials factor into linear factors (with integer coefficients). We express the product ${\nabla }_{D,C}{\nabla }_{B,A}$ as a $\mathbb{Z}$-linear combination of ${\nabla }_{U,V}$'s. More generally, for any two set compositions (i.e., ordered set partitions) $\mathbf{A}$ and $\mathbf{B}$ of $\left\{1,2,\dots ,n\right\}$, we define ${\nabla }_{\mathbf{B},\mathbf{A}}\in \mathcal{A}$ to be the sum of all permutations $w\in S_n$ that send each block of $\mathbf{A}$ to the corresponding block of $\mathbf{B}$. This generalizes ${\nabla }_{B,A}$. The factorization property of minimal polynomials does not extend to the ${\nabla }_{\mathbf{B},\mathbf{A}}$, but we describe the ideal spanned by the ${\nabla }_{\mathbf{B},\mathbf{A}}$ and a further ideal complementary to it. These two ideals have a "mutually annihilative" relationship, are free as $\mathbf{k}$-modules, and appear as annihilators of tensor product $S_n$-representations; they are also closely related to Murphy's cellular bases, Specht modules, pattern-avoiding permutations and even some algebras appearing in quantum information theory.

86 pages. Sections 1 and 2 are largely independent (and will probably become separate articles), except for the dimension of the span in Section 1 using Section 2