A refined count of coxeter element reflection factorizations

Elise Delmas; Thomas Hameister; Victor Reiner

doi:10.37236/7362

A refined count of coxeter element reflection factorizations

Elise Delmas, Thomas Hameister, Victor Reiner

Mathematics

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

7 Scopus citations

Abstract

For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the number of reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter number, expressed via data coming from a hyperplane orbit; a case-free proof of this due to J. Michel is included.

Original language	English (US)
Journal	Electronic Journal of Combinatorics
Volume	25
Issue number	1
DOIs	https://doi.org/10.37236/7362
State	Published - Feb 16 2018

Bibliographical note

Funding Information:
∗Supported by NSF grant DMS-1601961. †Supported by NSF grant DMS-1148634. ‡Supported by NSF grant DMS-1601961.

Funding Information:
Supported by NSF grant DMS-1601961. † Supported by NSF grant DMS-1148634. ‡ Supported by NSF grant DMS-1601961. Work of the second author was carried out under the auspices of the 2017 summer REU program at the School of Mathematics, University of Minnesota, Twin Cities. The authors thank Craig Corsi, Theo Douvropoulos, and Joel Lewis for helpful comments, and they thank Jean Michel for allowing them to include his proof of Corollary 2.

Publisher Copyright:
© 2018, Australian National University. All rights reserved.

Keywords

Coxeter element
Factorization
Reflection group
Well-generated

Access

10.37236/7362

OpenUrl availability

Full text

Cite this

@article{a7bb59468c7a40b5a420c717da08a13a,

title = "A refined count of coxeter element reflection factorizations",

abstract = "For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the number of reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter number, expressed via data coming from a hyperplane orbit; a case-free proof of this due to J. Michel is included.",

keywords = "Coxeter element, Factorization, Reflection group, Well-generated",

author = "Elise Delmas and Thomas Hameister and Victor Reiner",

note = "Funding Information: ∗Supported by NSF grant DMS-1601961. †Supported by NSF grant DMS-1148634. ‡Supported by NSF grant DMS-1601961. Funding Information: Supported by NSF grant DMS-1601961. † Supported by NSF grant DMS-1148634. ‡ Supported by NSF grant DMS-1601961. Work of the second author was carried out under the auspices of the 2017 summer REU program at the School of Mathematics, University of Minnesota, Twin Cities. The authors thank Craig Corsi, Theo Douvropoulos, and Joel Lewis for helpful comments, and they thank Jean Michel for allowing them to include his proof of Corollary 2. Publisher Copyright: {\textcopyright} 2018, Australian National University. All rights reserved.",

year = "2018",

month = feb,

day = "16",

doi = "10.37236/7362",

language = "English (US)",

volume = "25",

journal = "Electronic Journal of Combinatorics",

issn = "1077-8926",

publisher = "Electronic Journal of Combinatorics",

number = "1",

}

TY - JOUR

T1 - A refined count of coxeter element reflection factorizations

AU - Delmas, Elise

AU - Hameister, Thomas

AU - Reiner, Victor

N1 - Funding Information: ∗Supported by NSF grant DMS-1601961. †Supported by NSF grant DMS-1148634. ‡Supported by NSF grant DMS-1601961. Funding Information: Supported by NSF grant DMS-1601961. † Supported by NSF grant DMS-1148634. ‡ Supported by NSF grant DMS-1601961. Work of the second author was carried out under the auspices of the 2017 summer REU program at the School of Mathematics, University of Minnesota, Twin Cities. The authors thank Craig Corsi, Theo Douvropoulos, and Joel Lewis for helpful comments, and they thank Jean Michel for allowing them to include his proof of Corollary 2. Publisher Copyright: © 2018, Australian National University. All rights reserved.

PY - 2018/2/16

Y1 - 2018/2/16

N2 - For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the number of reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter number, expressed via data coming from a hyperplane orbit; a case-free proof of this due to J. Michel is included.

AB - For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the number of reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter number, expressed via data coming from a hyperplane orbit; a case-free proof of this due to J. Michel is included.

KW - Coxeter element

KW - Factorization

KW - Reflection group

KW - Well-generated

UR - http://www.scopus.com/inward/record.url?scp=85042237814&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042237814&partnerID=8YFLogxK

U2 - 10.37236/7362

DO - 10.37236/7362

M3 - Article

AN - SCOPUS:85042237814

SN - 1077-8926

VL - 25

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

IS - 1

ER -

A refined count of coxeter element reflection factorizations

Abstract

Bibliographical note

Keywords

Access

OpenUrl availability

Other files and links

Fingerprint

Cite this