Marked multi-colorings, partially commutative Lie superalgebras and right-angled Coxeter groups
- 1Department of Mathematics, Indian Institute of Science, Bangalore 560012 (India)
- 2Department of Mathematics, TUM School of Computation, Information and Technology, Technical University of Munich, Boltzmannstr. 3, 85748 Garching bei München (Germany)
Abstract
Infinite-dimensional Lie superalgebras, particularly Borcherds-Kac-Moody (BKM) superalgebras, play a fundamental role in mathematical physics, number theory, and representation theory. In this paper, we study the root multiplicities of BKM superalgebras via their denominator identities, deriving explicit combinatorial formulas in terms of graph invariants associated with marked (quasi) Dynkin diagrams. We introduce partially commutative Lie superalgebras (PCLSAs) and provide a direct combinatorial proof of their denominator identity, where the generating set runs over the super heaps monoid. A key notion in our approach is marked multi-colorings and their associated polynomials, which generalize chromatic polynomials and offer a method for computing root multiplicities. As applications, we characterize the roots of PCLSAs and establish connections between their universal enveloping algebras and right-angled Coxeter groups, leading to explicit formulas for their Hilbert series. These results further deepen the interplay between Lie superalgebras, graph theory, and algebraic combinatorics.
Related
- Commutative exact superalgebras and super-modular tensor categories
- Centroids Of Partially Commutative Two-step Nilpotent Groups
- Arithmetic growth in partially commutative semigroups of endomorphisms
- Poincaré duality for k-A Lie superalgebras
- Irreducibility of Verma modules for symmetrizable Kac-Moody superalgebras
BibTeX
@article{pilakkat2026marked,
title = {{Marked multi-colorings, partially commutative Lie superalgebras and right-angled Coxeter groups}},
author = {Chaithra Pilakkat and Deniz Kus and R. Venkatesh},
journal = {Algebraic Combinatorics},
year = {2026},
month = oct,
doi = {10.5802/alco.509},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.509/},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Marked multi-colorings, partially commutative Lie superalgebras and right-angled Coxeter groups
AU - Chaithra Pilakkat
AU - Deniz Kus
AU - R. Venkatesh
PY - 2026
DA - 2026/10/27
JO - Algebraic Combinatorics
DO - 10.5802/alco.509
UR - https://alco.centre-mersenne.org/articles/10.5802/alco.509/
AB - Infinite-dimensional Lie superalgebras, particularly Borcherds-Kac-Moody (BKM) superalgebras, play a fundamental role in mathematical physics, number theory, and representation theory. In this paper, we study the root multiplicities of BKM superalgebras via their denominator identities, deriving explicit combinatorial formulas in terms of graph invariants associated with marked (quasi) Dynkin diagrams. We introduce partially commutative Lie superalgebras (PCLSAs) and provide a direct combinatorial proof of their denominator identity, where the generating set runs over the super heaps monoid. A key notion in our approach is marked multi-colorings and their associated polynomials, which generalize chromatic polynomials and offer a method for computing root multiplicities. As applications, we characterize the roots of PCLSAs and establish connections between their universal enveloping algebras and right-angled Coxeter groups, leading to explicit formulas for their Hilbert series. These results further deepen the interplay between Lie superalgebras, graph theory, and algebraic combinatorics.
SN - 2589-5486
ER -