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A Type B analog of the Whitehouse representation

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Abstract

We give a Type B analog of Whitehouse’s lifts of the Eulerian representations from \(S_n\) to \(S_{n+1}\) by introducing a family of \(B_{n}\)-representations that lift to \(B_{n+1}\). As in Type A, we interpret these representations combinatorially via a family of orthogonal idempotents in the Mantaci-Reutenauer algebra, and topologically as the graded pieces of the cohomology of a certain \({{\,\mathrm{\mathbb {Z}}\,}}_{2}\)-orbit configuration space of \({{\,\mathrm{\mathbb {R}}\,}}^{3}\). We show that the lifted \(B_{n+1}\)-representations also have a configuration space interpretation, and further parallel the Type A story by giving analogs of many of its notable properties, such as connections to equivariant cohomology and the Varchenko-Gelfand ring.

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Notes

  1. In fact, in [9], Bergeron–Bergeron–Howlett–Taylor defined analogous idempotents for any finite Coxeter group.

  2. The word hidden refers to the fact that there is not an obvious \(S_{n+1}\) action on \({{\,\textrm{Conf}\,}}_n({{\,\mathrm{\mathbb {R}}\,}}^{d})\).

  3. The notation \({{\,\textrm{in}\,}}_{\deg }({\mathcal {Q}})\) refers to the fact that \({{\,\textrm{in}\,}}_{\deg }({\mathcal {Q}})\) is an initial ideal for some degree (partial) ordering on monomials. This will be rigorously defined and discussed in Sect. 3.1.

  4. In fact (2.1.4) holds for any odd \(d \ge 3\) by replacing \(H^{2k}{{\,\textrm{Conf}\,}}_{n}({{\,\mathrm{\mathbb {R}}\,}}^{d})\) with \(H^{(d-1)k}{{\,\textrm{Conf}\,}}_{n}({{\,\mathrm{\mathbb {R}}\,}}^{d})\).

  5. We think of this as a recursion in the sense that the formula relates the representation \(E_{n}^{(k)}\) (which lifts to \(F_{n+1}^{(k)}\)) to the representation \(F_{n}^{(k)}\) (which restricts to \(E_{n-1}^{(k)}\)).

  6. The terminology of “hands” and “fingers” originates in Hélène Barcelo’s thesis [5, Thm 2.1] and was later used in Barcelo–Goupil [7], both in the context of describing an nbc-basis. Such bases arise in the study of matroids. While we are not in the matroid (e.g. hyperplane) setting, because our description of \({\mathcal {N}}\) uses the hand/finger description, we may refer to it basis as an nbc-basis nonetheless.

  7. In [24], Feichtner–Ziegler give a presentation of \(H^{*}\mathcal {Z}_{n}^d\) for \(d \ge 2\). However, their computation of the action of \(B_n\) on the generators of \(H^{*}\mathcal {Z}_{n}^d\) has an error [24, Lemma 7(iv)]. Xicoténcatl also gives a presentation of \(H^{*}\mathcal {Z}_{n}^d\), which agrees with our presentation; however his work does not explicitly compute the action of \(B_n\) on the generators of \(H^{*}\mathcal {Z}_{n}^d\). We will see in Proposition 4.6 that the \(B_n\)-action on \(H^{*}\mathcal {Z}_{n}^d\) is delicate, and so we include all the details of our computations for completeness.

  8. They further prove (with a bit more work) that the \(d=2\) case is also torsion free and satisfies (4.1.2).

  9. The proof of Theorem 5.10 does not rely on the remaining results in § 4. In particular, all subsequent work in §4 will focus on computing the presentation of \(H^{*}\mathcal {Z}_{n}^3\). Theorem 5.10 will be proved using equivariant formality, which we will see follows from the Hilbert series for \(H^{*}\mathcal {Z}_{n}^3\) given in (4.1.2).

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Acknowledgements

The author is very grateful to Vic Reiner for guidance and encouragement at every stage in this project, to Sheila Sundaram for insightful questions that served as the initial inspiration for this work, to Monica Vazirani for sharing her undergraduate thesis, and to Marcelo Aguiar, François Bergeron, Patty Commins, Christophe Hohlweg, Allen Knutson, Nick Proudfoot, Franco Saliola and Dev Sinha for helpful discussions. The author is supported by the NSF Graduate Research Fellowship (Award Number DMS-0007404).

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    Sarah Brauner

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Brauner, S. A Type B analog of the Whitehouse representation. Math. Z. 303, 58 (2023). https://doi.org/10.1007/s00209-022-03200-7

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