A pattern avoidance criterion for free inversion arrangements |
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Authors: | William Slofstra |
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Institution: | 1.Institute for Quantum Computing,University of Waterloo,Waterloo,Canada |
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Abstract: | We show that the hyperplane arrangement of a coconvex set in a finite root system is free if and only if it is free in corank 4. As a consequence, we show that the inversion arrangement of a Weyl group element w is free if and only if w avoids a finite list of root system patterns. As a key part of the proof, we use a recent theorem of Abe and Yoshinaga to show that if the root system does not contain any factors of type C or F, then Peterson translation of coconvex sets preserves freeness. This also allows us to give a Kostant–Shapiro–Steinberg rule for the coexponents of a free inversion arrangement in any type. |
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