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A new statistic on the hyperoctahedral groups

A new statistic on the hyperoctahedral groups
A new statistic on the hyperoctahedral groups
We introduce a new statistic on the hyperoctahedral groups (Coxeter groups of type B), and give a conjectural formula for its signed distributions over arbitrary descent classes. The statistic is analogous to the classical Coxeter length function, and features a parity condition. For descent classes which are singletons the conjectured formula gives the Poincaré polynomials of the varieties of symmetric matrices of fixed rank.

For several descent classes we prove the conjectural formula. For this we construct suitable supporting sets for the relevant generating functions. We prove cancellations on the complements of these supporting sets using suitably defined sign reversing
involutions.
hyperoctahedral groups, signed permutation statistics, sign reversing involutions, descent sets, generating functions
1077-8926
1-23
Stasinski, Alexander
94bd8be7-4b4f-4e22-875b-3628d8c2ca19
Voll, Christopher
b7bd2890-38ac-4e05-adb7-3b376916ff79
Stasinski, Alexander
94bd8be7-4b4f-4e22-875b-3628d8c2ca19
Voll, Christopher
b7bd2890-38ac-4e05-adb7-3b376916ff79

Stasinski, Alexander and Voll, Christopher (2013) A new statistic on the hyperoctahedral groups. The Electronic Journal of Combinatorics, 20 (3), 1-23.

Record type: Article

Abstract

We introduce a new statistic on the hyperoctahedral groups (Coxeter groups of type B), and give a conjectural formula for its signed distributions over arbitrary descent classes. The statistic is analogous to the classical Coxeter length function, and features a parity condition. For descent classes which are singletons the conjectured formula gives the Poincaré polynomials of the varieties of symmetric matrices of fixed rank.

For several descent classes we prove the conjectural formula. For this we construct suitable supporting sets for the relevant generating functions. We prove cancellations on the complements of these supporting sets using suitably defined sign reversing
involutions.

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More information

Published date: 26 September 2013
Keywords: hyperoctahedral groups, signed permutation statistics, sign reversing involutions, descent sets, generating functions
Organisations: Mathematical Sciences

Identifiers

Local EPrints ID: 366839
URI: https://eprints.soton.ac.uk/id/eprint/366839
ISSN: 1077-8926
PURE UUID: d5908979-c5b4-4b59-bd1c-ab8ec1c2a54c

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Date deposited: 11 Jul 2014 14:39
Last modified: 05 Oct 2018 12:09

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Contributors

Author: Alexander Stasinski
Author: Christopher Voll

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