Abstract
We describe a type B analog of the much studied Lie representation of the symmetric group. The nth Lie representation of Sn restricts to the regular representation of Sn−1, and our generalization mimics this property. Specifically, we construct a representation of the type B Weyl group Bn that restricts to the regular representation of Bn−1. We view both of these representations as coming from the internal zonotopal algebra of the Gale dual of the corresponding reflection arrangements.
Highlights
We consider the polynomial ring S in two sets of variables yij and zij, where 1 ≤ i < j ≤ n
We have the following conventions on the indices: For all 1 ≤ i, j ≤ n with i = j, yji = −yij and zji = zij. This ring has an action of the Weyl group of type B, which we denote by Bn
We propose the top degree component of the quotient S/J as a type B analog of the Lie representation of the symmetric group Sn, Lien
Summary
We consider the polynomial ring S in two sets of variables yij and zij, where 1 ≤ i < j ≤ n. The Hilbert series of the Orlik-Solomon algebra of the type B reflection arrangement is (1 + q)(1 + 3q) · · · (1 + (2n − 1)q) and its top degree component has dimension 1 · 3 · 5 · · · (2n − 1) This is the dimension of the whole quotient occurring in Theorem 1.1, not a single graded piece. The quotient ring of Theorem 1.1 is the so-called internal zonotopal algebra of the type B reflection arrangement These algebras were studied by Holtz and Ron [HR], where connections to box splines are made, and later by Ardila and Postnikov [AP10, AP15], among others. We motivate Theorem 1.1 by studying the internal zonotopal algebra of the Gale dual of the braid arrangement This makes a connection to the well-studied Whitehouse representation of the symmetric group through work of Mathieu [Ma], Gaiffi [Ga], and Robinson and Whitehouse [RW]. We propose the study of decreasing ±trees, which are a type B analog of decreasing trees
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