arXiv subjects
Publications and source records attributed to Himadri Mukherjee.
In a finite distributive lattice $Ł$ we define two functions $s(α)=|\{δ\in \mathcal{L} | δ\leq α\}|$ and $l(α)=|\{δ\in \mathcal{L} | δ\geq α\}|$. In this present article we prove that the sum of these two functions over a finite distributive lattice are equal. Using this identity we give a formula for the number of non-comparable pairs of elements in a finite distributive lattice.
Toric varieties associated with distributive lattices arise as a fibre of a flat degeneration of a Schubert variety in a minuscule. The singular locus of these varieties has been studied by various authors. In this article we prove that the number of diamonds incident on a lattice point $\a$ in a product of chain lattices is more than or equal to the codimension of the lattice. Using this we also show that the lattice varieties associated with product of chain lattices is smooth.