$${{\,\mathrm{\mathbb {P}GL}\,}}$$ Orbits in Tree Varieties
Year: 2025
Type: article
Abstract: In this paper, we introduce tree varieties as a natural generalization of products of partial flag varieties. We study orbits of the $${{\,\mathrm{\mathbb {P}GL}\,}}$$ action on tree varieties. We characterize tree varieties with finitely many $${{\,\mathrm{\mathbb {P}GL}\,}}$$ orbits, generalizing a celebrated theorem of Magyar, Weyman and Zelevinsky. We give criteria that guarantee that a tree variety has a dense $${{\,\mathrm{\mathbb {P}GL}\,}}$$ orbit and provide many examples of tree varieties that do not have dense $${{\,\mathrm{\mathbb {P}GL}\,}}$$ orbits. We show that a triple of two-step flag varieties $$F(k_1, k_2; n)^3$$ has a dense $${{\,\mathrm{\mathbb {P}GL}\,}}(n)$$ orbit if and only if $$k_1 + k_2 \not = n$$ .
Source: Transformation Groups
Authors: Izzet Coskun, Demir Eken, Chris Yun
Institutions: University of Illinois Chicago, University of Michigan
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
Awards: G340925115, G7203542996
OpenAlex ID: W4413873116