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 $n\geq3$, 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 $\log\dim C_n\geq(1-o(1))n^5(\log n)^2$ as $n\to\infty$. 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.