A Pair of Garside Shadows
Piotr Przytycki, Yeeka Yau
TL;DR
This paper proves the existence of minimal elements in certain algebraic structures called Shi parts and cone type parts, showing they form Garside shadows, thus resolving multiple conjectures in the field.
Contribution
It establishes that the smallest elements in Shi parts and cone type parts form Garside shadows, confirming conjectures by Parkinson, Hohlweg, Nadeau, and Williams.
Findings
Existence of smallest elements in Shi parts and cone type parts.
These elements form Garside shadows.
Resolution of multiple longstanding conjectures.
Abstract
We prove that the smallest elements of Shi parts and cone type parts exist and form Garside shadows. The latter resolves a conjecture of Parkinson and the second author as well as a conjecture of Hohlweg, Nadeau and Williams.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometric and Algebraic Topology · semigroups and automata theory