Seminar: Graphs of Reduced Words and Some Connections
Speaker: Praise Adeyemo, University of Ibadan, Nigeria
Abstract: It is well-known that the Weyl group $W$ of the general linear group $\text{GL}_{n}(\mathbb{C})$ is isomorphic to the group $S_n$ of permutations. The family of graphs of reduced words for permutations of the form $$[n, 1, 2, 3, . . . , n − 4, n − 1, n − 3]\text{, for } n \geq 4$$ is investigated. These permutations are characterised by the hook cycle type $(n − 2, 1, 1)$. The closed formula obtained for the number of reduced words reveals an intereting connection to the set $W^{n−2}_{P_{n−2}}$ of minimal length left coset representatives of the coset space $W/W_{P_{n−2}}$. These are precisely the Grassmannian permutations for the Grassmannian $\text{Gr}(2, n)$. These in turn, are connected to the lattice points $(n − 2)\Delta_2 \cap \mathbb{Z}^{2}_{\geq 0}$ of the $(n − 2)$-dilation of the standard-$2$-simplex. It turns out that the Hilbert series of the integral cohomology ring of the Grassmannian $\text{Gr}(2, n)$ coincides with the refinement $\mathsf{L}^\mathsf{a}_{\Delta_2} (n − 2)$ of the Ehrhart polynomial of the standard two simplex, where $\mathsf{a}$ is the weight $(1, 2)$.