LS algebras and application to Schubert varieties
The author defines LS algebras using LS paths. Given a finite partially ordered set \((S,\leq)\) and a function \(f\), such that for \(a<b\) from \(S\), \(f(a,b)\) is a natural number. Then an LS path of degree \(r\) is defined to be a pair of a path \(a_1<a_2< \cdots< a_s\) and a set of rational numbers \(0= x_0< x_1<\cdots <x_s=r\), such that for any \(i\), \(x_i\cdot f(a_i,a_{i+1}) \in \mathbb Z\) (thus the denominators are limited by \(f(a_i,a_{i+1})\).) Let \(L_r\) be the set of LS paths of degree \(r\), and then the multiplication is defined on \(R \oplus RL_1\oplus RL_2\oplus \dots\) to get an LS algebra over a ring \(R\). The Stanley-Reisner ring of \((S,\leq)\) is a particular case of the construction for \(f\equiv 1\). The author studies deformations of LS-algebras and uses the results to prove the main theorem: Any Schubert variety admits a flat deformation to a union of normal toric varieties whose irreducible components are indexed by the maximal chains of a special poset.
scientific article; zbMATH DE number 1530009
scientific article