A new bijective proof of the q-Pfaff–Saalschütz identity with applications to quantum groups

Álvaro Gutierrez Caceres; Álvaro Martinez; Michal Szwej; Mark J Wildon

doi:10.48550/arXiv.2505.08422

A new bijective proof of the q-Pfaff–Saalschütz identity with applications to quantum groups

Álvaro Gutierrez Caceres, Álvaro Martinez, Michal Szwej, Mark J Wildon

Research output: Contribution to journal › Article (Academic Journal) › peer-review

Abstract

We present a combinatorial proof of the q-Pfaff–Saalschutz identity by a composition of explicit bijections, in which q-binomial coefficients are interpreted as counting subspaces of Fq-vector spaces. As a corollary, we obtain
a new multiplication rule for quantum binomial coefficients and hence a new presentation of Lusztig’s integral form UZ[q,q−1] (sl2) of the Cartan subalgebra of the quantum group Uq(sl2).

Original language	English
Article number	104321
Number of pages	15
Journal	European Journal of Combinatorics
Volume	133
Early online date	22 Dec 2025
DOIs	https://doi.org/10.48550/arXiv.2505.08422 https://doi.org/10.1016/j.ejc.2025.104321
Publication status	Published - 1 Mar 2026

Bibliographical note

Publisher Copyright:
© 2025 The Authors.

Access to Document

10.48550/arXiv.2505.08422Licence: CC BY-NC-ND
10.1016/j.ejc.2025.104321Licence: CC BY-NC-ND

Handle.net

Persistent link

Cite this

@article{a63645296f724d149eb3ff173a06ece2,

title = "A new bijective proof of the q-Pfaff–Saalsch{\"u}tz identity with applications to quantum groups",

abstract = "We present a combinatorial proof of the q-Pfaff–Saalschutz identity by a composition of explicit bijections, in which q-binomial coefficients are interpreted as counting subspaces of Fq-vector spaces. As a corollary, we obtain a new multiplication rule for quantum binomial coefficients and hence a new presentation of Lusztig{\textquoteright}s integral form UZ[q,q−1] (sl2) of the Cartan subalgebra of the quantum group Uq(sl2).",

author = "\{Gutierrez Caceres\}, {\'A}lvaro and {\'A}lvaro Martinez and Michal Szwej and Wildon, \{Mark J\}",

note = "Publisher Copyright: {\textcopyright} 2025 The Authors. ",

year = "2026",

month = mar,

day = "1",

doi = "10.48550/arXiv.2505.08422",

language = "English",

volume = "133",

journal = "European Journal of Combinatorics",

issn = "0195-6698",

publisher = "Academic Press",

}

TY - JOUR

T1 - A new bijective proof of the q-Pfaff–Saalschütz identity with applications to quantum groups

AU - Gutierrez Caceres, Álvaro

AU - Martinez, Álvaro

AU - Szwej, Michal

AU - Wildon, Mark J

PY - 2026/3/1

Y1 - 2026/3/1

N2 - We present a combinatorial proof of the q-Pfaff–Saalschutz identity by a composition of explicit bijections, in which q-binomial coefficients are interpreted as counting subspaces of Fq-vector spaces. As a corollary, we obtain a new multiplication rule for quantum binomial coefficients and hence a new presentation of Lusztig’s integral form UZ[q,q−1] (sl2) of the Cartan subalgebra of the quantum group Uq(sl2).

AB - We present a combinatorial proof of the q-Pfaff–Saalschutz identity by a composition of explicit bijections, in which q-binomial coefficients are interpreted as counting subspaces of Fq-vector spaces. As a corollary, we obtain a new multiplication rule for quantum binomial coefficients and hence a new presentation of Lusztig’s integral form UZ[q,q−1] (sl2) of the Cartan subalgebra of the quantum group Uq(sl2).

U2 - 10.48550/arXiv.2505.08422

DO - 10.48550/arXiv.2505.08422

M3 - Article (Academic Journal)

SN - 0195-6698

VL - 133

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

M1 - 104321

ER -

A new bijective proof of the q-Pfaff–Saalschütz identity with applications to quantum groups

Abstract

Bibliographical note

Access to Document

Handle.net

Fingerprint

Cite this