Finite noncommutative coinvariant algebras
Abstract
We study quotients of free associative algebras by the two-sided ideals generated by positive-degree symmetric, quasisymmetric, or free quasisymmetric polynomials. Free quasisymmetric functions give a word realization of the Malvenuto-Reutenauer Hopf algebra of permutations. For quotients on n variables, finiteness in the symmetric and quasisymmetric cases follows from a theorem of Bergman--Isaacs when n! is invertible in the field. Our main result is a nilpotence criterion for families of relations compatible with restriction to subalphabets. The criterion proves finiteness over every field for quasisymmetric, r-quasisymmetric, and free quasisymmetric coinvariants, with explicit degree bounds exponential in n for fixed r. In the quasisymmetric and r-quasisymmetric cases, these improve the bounds inherited from Bergman--Isaacs. A principal application is to the free quasisymmetric case, where, for n≥3, the group-invariant argument used for the symmetric case does not apply. For the symmetric quotient Cn, we prove that invertibility of n! is also necessary for finiteness. In characteristic zero, an auxiliary presentation and the Golod--Shafarevich inequality give logdim Cn≥(1−o(1))n5(log n)2 as n→∞. The same inequality shows that quasisymmetric and free quasisymmetric coinvariant dimensions grow faster than every fixed power of n!. For all three families on several freely noncommuting alphabets, we determine all multigraded Hilbert series and, for symmetric coinvariants in characteristic zero, Frobenius characteristics from one-alphabet data. In characteristic zero, projection gives stable Schur multiplicities for compatible quotients in mixed commuting, anticommuting, and free alphabets.
Related
- Derivation based noncommutative geometry of quiver path algebras
- Equivariant Morita equivalence for finite cyclic group actions on noncommutative tori
- The bigraded Frobenius series of the wreath product superspace coinvariant rings
- Keller's conjecture for split finite-dimensional algebras
- The Cartan determinant conjecture for representation-finite Artin algebras
BibTeX
@misc{aliniaeifard2026finite,
title = {{Finite noncommutative coinvariant algebras}},
author = {Farid Aliniaeifard and Nantel Bergeron and Huilan Li and Shu Xiao Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16873},
eprint = {2610.16873},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16873},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Finite noncommutative coinvariant algebras
AU - Farid Aliniaeifard
AU - Nantel Bergeron
AU - Huilan Li
AU - Shu Xiao Li
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16873
UR - https://arxiv.org/abs/2610.16873
AB - We study quotients of free associative algebras by the two-sided ideals generated by positive-degree symmetric, quasisymmetric, or free quasisymmetric polynomials. Free quasisymmetric functions give a word realization of the Malvenuto-Reutenauer Hopf algebra of permutations. For quotients on n variables, finiteness in the symmetric and quasisymmetric cases follows from a theorem of Bergman--Isaacs when n! is invertible in the field. Our main result is a nilpotence criterion for families of relations compatible with restriction to subalphabets. The criterion proves finiteness over every field for quasisymmetric, r-quasisymmetric, and free quasisymmetric coinvariants, with explicit degree bounds exponential in n for fixed r. In the quasisymmetric and r-quasisymmetric cases, these improve the bounds inherited from Bergman--Isaacs. A principal application is to the free quasisymmetric case, where, for n≥3, the group-invariant argument used for the symmetric case does not apply. For the symmetric quotient C_n, we prove that invertibility of n! is also necessary for finiteness. In characteristic zero, an auxiliary presentation and the Golod--Shafarevich inequality give logdim C_n≥(1−o(1))n^5(log n)^2 as n→∞. The same inequality shows that quasisymmetric and free quasisymmetric coinvariant dimensions grow faster than every fixed power of n!. For all three families on several freely noncommuting alphabets, we determine all multigraded Hilbert series and, for symmetric coinvariants in characteristic zero, Frobenius characteristics from one-alphabet data. In characteristic zero, projection gives stable Schur multiplicities for compatible quotients in mixed commuting, anticommuting, and free alphabets.
ER -