Christian Smith, NC State, The Algebra of “up-operators” for Young’s Lattice and Bruhat Order on S_n

September 27, 2019 | 11:45 am - 12:45 pm EDT

Let $U$ be a free associative algebra over $\mathbb{C}$ generated by $u_j$ for $j$ in some indexing set $J$ and let $P$ be a poset. For $u_j \in U$ and $p \in P$ we define an action of $U$ on $\mathbb{C}[P]$ (the complex vector space with basis $P$ ) in a way such that $u_j$ either annihilates $p$ or sends it to $p' \in P$ where $p'$ covers $p$ and we extend multiplicatively and linearly. Let $I$ be the two-sided ideal which annihilates all elements of $\mathbb{C}[P]$ . We characterize $U / I$ when $P$ is Young’s Lattice and we discuss the case when $P$ is the Bruhat Order on $S_n$ .