LS algebras and application to Schubert varieties
type: Item
instance of: scholarly article
title: LS algebras and application to Schubert varieties
zbMATH Open document ID: 1019.14019
DOI: 10.1007/BF01679715
published in: Transformation Groups
publication date: 2001-10-07T00:00:00Z
review text: 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.
reviewed by: Alexei Rudakov
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 06A11
zbMATH DE Number: 1530009
zbMATH Keywords: LS algebras
zbMATH Keywords: LS paths
zbMATH Keywords: Stanley-Reisner ring
zbMATH Keywords: Schubert variety
zbMATH Keywords: toric varieties
author: Rocco Chirivì
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1530009