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Schubert varieties, inversion arrangements, and Peterson translation

Abstract : We show that an element $\mathcal{w}$ of a finite Weyl group W is rationally smooth if and only if the hyperplane arrangement $\mathcal{I} (\mathcal{w})$ associated to the inversion set of \mathcal{w} is inductively free, and the product $(d_1+1) ...(d_l+1)$ of the coexponents $d_1,\ldots,d_l$ is equal to the size of the Bruhat interval [e,w]. We also use Peterson translation of coconvex sets to give a Shapiro-Steinberg-Kostant rule for the exponents of $\mathcal{w}$.
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William Slofstra. Schubert varieties, inversion arrangements, and Peterson translation. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.715-726. ⟨hal-01207554⟩

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