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- W3163994809 abstract "Given a finite crystallographic root system $Phi$ whose Dynkin diagram has a non-trivial automorphism, it yields a new root system $Phi_{tau}$ by a so-called classical folding. On the other hand, Lusztig's folding (1983) folds the root system of type $E_8$ to $H_4$ starting from an automorphism of the root lattice of type $E_8.$ The notion of a twisted quadratic folding of a root system was introduced by Lanini-Zainoulline (2018) to describe both the classical foldings and Lusztig's folding on the same footing. The structure algebra $mathcal{Z}(mathcal{G})$ of the moment graph $mathcal{G}$ associated with a finite root system and its reflection group $W$ is an algebra over a certain polynomial ring $mathcal{S},$ whose underlying module is free with a distinguished basis ${sigma^{(w)} mid w in W}$ called combinatorial Schubert classes. By Lanini-Zainoulline (2018), a twisted quadratic folding $Phi rightsquigarrow Phi_{tau}$ induces an embedding of the respective Coxeter groups $varepsilon: W_{tau} hookrightarrow W$ and a ring homomorphism $varepsilon^*: mathcal{Z}(mathcal{G}) rightarrow mathcal{Z}(mathcal{G}_{tau})$ between the corresponding structure algebras. This paper studies the $varepsilon^*$-preimage of Schubert classes and provides a combinatorial criterion for a Schubert class $sigma^{(u)}_{tau}$ of $mathcal{Z}(mathcal{G}_{tau})$ to admit a Schubert class $sigma^{(w)}$ of $mathcal{Z}(mathcal{G})$ such that the relation $varepsilon^*(sigma^{(w)}) = c cdot sigma^{(u)}_{tau}$ holds for some nonzero scalar $c.$" @default.
- W3163994809 created "2021-06-07" @default.
- W3163994809 creator A5054548759 @default.
- W3163994809 date "2021-05-26" @default.
- W3163994809 modified "2023-09-27" @default.
- W3163994809 title "Twisted quadratic foldings of root systems and liftings of Schubert classes" @default.
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