The Worpitzky identity for the groups of signed and even-signed permutations

Eli Bagno; David Garber; Mordechai Novick

doi:10.1007/s10801-021-01056-4

The Worpitzky identity for the groups of signed and even-signed permutations

Eli Bagno, David Garber, Mordechai Novick

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

The well-known Worpitzky identity (x+1)n=∑k=0n-1An,k(x+n-kn)provides a connection between two bases of Q[x] : the standard basis (x+ 1) ⁿ and the binomial basis (x+n-kn), where the Eulerian numbers A_n_,_k for the symmetric group serve as the entries of the transformation matrix. Brenti has generalized this identity to the Coxeter groups of types B_n and D_n (signed and even-signed permutations groups, respectively) using generatingfunctionology. Motivated by Foata–Schützenberger’s and Rawlings’ proof for the Worpitzky identity in the symmetric group, we provide combinatorial proofs for the generalizations of this identity and for their q-analogues in the Coxeter groups of types B_n and D_n. Our proofs utilize the language of P-partitions for the B_n- and D_n-posets, introduced by Chow and Stembridge, respectively.

Original language	English
Pages (from-to)	413-428
Number of pages	16
Journal	Journal of Algebraic Combinatorics
Volume	55
Issue number	2
DOIs	https://doi.org/10.1007/s10801-021-01056-4
State	Published - Mar 2022
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Funding

Eli Bagno and David Garber were partially supported by the Israeli Ministry of Science and Technology, and the French National Scientific Research Center (CNRS), Grant PRC 1656.

Funders	Funder number
French National Scientific Research Center
Centre National de la Recherche Scientifique	PRC 1656
Ministry of science and technology, Israel

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abstract = "The well-known Worpitzky identity (x+1)n=∑k=0n-1An,k(x+n-kn)provides a connection between two bases of Q[x] : the standard basis (x+ 1) n and the binomial basis (x+n-kn), where the Eulerian numbers An,k for the symmetric group serve as the entries of the transformation matrix. Brenti has generalized this identity to the Coxeter groups of types Bn and Dn (signed and even-signed permutations groups, respectively) using generatingfunctionology. Motivated by Foata–Sch{\"u}tzenberger{\textquoteright}s and Rawlings{\textquoteright} proof for the Worpitzky identity in the symmetric group, we provide combinatorial proofs for the generalizations of this identity and for their q-analogues in the Coxeter groups of types Bn and Dn. Our proofs utilize the language of P-partitions for the Bn- and Dn-posets, introduced by Chow and Stembridge, respectively.",

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The Worpitzky identity for the groups of signed and even-signed permutations

Abstract

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