Subroot systems and total positivity in finite reflection groups

Abstract

Given a root system R R and the corresponding finite reflection group W W let Hom ⁡ ( W , Z ^ 2 ) \operatorname {Hom}(W,\,\widehat {\mathbb Z}_2) be the group of homomorphisms from W W into Z ^ 2 \widehat {\mathbb Z}_2 , where Z ^ 2 = { 1 , − 1 } \widehat {\mathbb Z}_2=\{1,-1\} with multiplication. We propose a procedure of constructing subroot systems of R R by using homomorphisms η ∈ Hom ⁡ ( W , Z ^ 2 ) \eta \in \operatorname {Hom}(W,\,\widehat {\mathbb Z}_2) . This construction is next used for establishing a relation between concepts of total positivity and η \eta -total positivity.