Injectivity of the canonical map from the universal algebra of pseudo-roots $Q_n$ into the free skew field

The algebra $Q_n$ of pseudo-roots of a generic noncommutative polynomial, introduced by Gelfand, Retakh and Wilson and studied further with S. Gelfand and Serconek, is the quadratic algebra on generators $x_{A,i}$ modulo the universal diamond relations satisfied by pseudo-roots, designed as an affordable intermediate between the free associative algebra and the free skew field. Its Hilbert series was computed in 2002, and the companion question of that paper, the Koszulity of $Q_n$, was proved affirmatively by Serconek-Wilson and independently by Piontkovski by 2005. What remains open is whether the canonical homomorphism $e\colon Q_n\to\mathcal{C}$ into the free skew field, sending each generator to the actual pseudo-root expressed through Vandermonde quasideterminants, is injective: this is known only for $n\le3$ (Gelfand-Retakh-Wilson 2001; the Quasideterminants survey 2005), and the general case is open. The problem is to prove or disprove that $e$ is injective for every $n$, equivalently, whether the pseudo-roots of a generic degree-$n$ polynomial in the free skew field satisfy exactly the diamond relations.

Background

Let $k$ be a field. The free associative algebra $k\langle x_1,\dots,x_n\rangle$ on $n$ noncommuting generators admits a universal field of fractions, the free skew field $\mathcal{C}=k\langle\langle x_1,\dots,x_n\rangle\rangle$ constructed by P. M. Cohn: it is, in a precise sense, the most general division ring generated by $x_1,\dots,x_n$, and every rational expression over the $x_i$ that is invertible somewhere can be interpreted in it.

Fix a central indeterminate $t$ and a monic polynomial $P(t)=t^n+a_1t^{n-1}+\dots+a_n$ with coefficients in an associative unital algebra $R$. Over a commutative coefficient ring, factorizations of $P$ into linear factors differ only by the order of the factors; over a noncommutative ring there can be many essentially different ones. An element $y\in R$ is a right root of $P$ if $P(t)$ is right-divisible by $t-y$, and a pseudo-root of $P$ if $P(t)=q_1(t)(t-y)q_2(t)$ for some $q_1,q_2\in R[t]$; pseudo-roots interpolate between the roots appearing in different factorizations of the same polynomial. When $x_1,\dots,x_n\in R$ form a generic set of right roots of $P$ (meaning every Vandermonde submatrix in the $x_i$ is invertible), a noncommutative analogue of Viete's theorem due to Gelfand and Retakh, based on the theory of quasideterminants, produces from every ordering $(i_1,\dots,i_n)$ a full factorization

$$P(t)=(t-x_{A_{n-1},i_n})(t-x_{A_{n-2},i_{n-1}})\cdots(t-x_{A_0,i_1})$$

with $A_j=\{i_1,\dots,i_j\}$, and expresses the coefficients $a_m$ as the elementary symmetric functions of the factors; this is the starting point of the theory of noncommutative symmetric functions. Here a quasideterminant is the noncommutative substitute for the determinant of a matrix over a division ring (the $(i,j)$-quasideterminant is the reciprocal of the $(i,j)$-entry of the inverse matrix), and the pseudo-root attached to a subset $A=\{i_1,\dots,i_k\}$ and an index $i\notin A$ is the conjugate

$$x_{A,i}=V(x_{i_1},\dots,x_{i_k},x_i)\,x_i\,V(x_{i_1},\dots,x_{i_k},x_i)^{-1}$$

of $x_i$, where $V(y_1,\dots,y_m)$ denotes the Vandermonde quasideterminant, the $(1,m)$-quasideterminant of the Vandermonde matrix in $y_1,\dots,y_m$.

The pseudo-roots of any such $P$ satisfy universal quadratic-linear identities, the diamond relations

$$x_{A\cup\{i\},j}+x_{A,i}=x_{A\cup\{j\},i}+x_{A,j},\qquad x_{A\cup\{i\},j}\,x_{A,i}=x_{A\cup\{j\},i}\,x_{A,j}$$

for $i\neq j$ outside $A$, equivalently $(t-x_{A\cup\{i\},j})(t-x_{A,i})=(t-x_{A\cup\{j\},i})(t-x_{A,j})$; these encode the independence of the displayed factorization of the chosen ordering. Gelfand, Retakh and Wilson therefore defined the universal algebra of pseudo-roots $Q_n$: the algebra over $k$ with one generator $x_{A,i}$ for each pair $(A,i)$ with $A\subseteq\{1,\dots,n\}$ and $i\notin A$, modulo the diamond relations. Their stated motivation is economic: addition and multiplication are cheap operations while division is expensive, so one wants an algebra strictly between the free associative algebra, which contains no pseudo-roots beyond the roots themselves, and the gigantic free skew field; $Q_n$ is designed to be that affordable intermediate.

The structural theory of $Q_n$ is now rather complete. It is a quadratic algebra (in the alternative generators $r_A$, indexed by the nonempty subsets $A$, the relations become homogeneous quadratic), and Gelfand, Gelfand, Retakh, Serconek and Wilson computed its Hilbert series, $H(Q_n,t)=(1-t)/(1-t(2-t)^n)$, where the Hilbert series of a graded algebra is the generating function of the dimensions of its graded components, as well as that of its quadratic dual $Q_n^!$ (the algebra on the dual space of generators with the orthogonally complementary relations), which is finite-dimensional with $H(Q_n^!,t)=(1+t(2+t)^n)/(1+t)$. The numerically necessary Koszulity condition $H(Q_n,t)H(Q_n^!,-t)=1$ held, and the question whether $Q_n$ is a Koszul algebra (a quadratic algebra is Koszul when the trivial module $k$ admits a linear free resolution) was settled affirmatively by Serconek and Wilson and, independently, by Piontkovski: $Q_n$ is Koszul of global dimension $n$ for every $n\ge2$. In addition, $Q_n$ has an explicit normal-form basis of monomials in the $r_A$, each of the $n!$ subalgebras generated by the pseudo-roots along one ordering is free on $n$ generators, and the intersection of all these subalgebras is precisely the algebra of noncommutative symmetric functions, a noncommutative analogue of the classical description of symmetric functions.

What remains unresolved is whether the abstract algebra $Q_n$ faithfully models actual pseudo-roots. Specializing the construction to the free skew field itself, the elements

$$V(x_{i_1},\dots,x_{i_k},x_i)\,x_i\,V(x_{i_1},\dots,x_{i_k},x_i)^{-1}$$

satisfy the diamond relations, so there is a canonical homomorphism $e\colon Q_n\to\mathcal{C}$ sending the generator $x_{A,i}$ to the actual pseudo-root, with image the subalgebra $\hat Q_n\subseteq\mathcal{C}$ generated by the pseudo-roots of a generic polynomial of degree $n$. Gelfand, Retakh and Wilson proved that $e$ is injective for $n=2$, and the Quasideterminants survey records the additional case $n=3$; the general statement was published by its authors as a conjecture and echoed in the 2002 Hilbert-series paper as a belief that $e$ is an embedding. That conjecture is the problem below.

Problem Statement

Prove or disprove: the canonical homomorphism

$$e\colon Q_n\longrightarrow\mathcal{C}$$

from the universal algebra of pseudo-roots $Q_n$ (the algebra over a field $k$ with generators $x_{A,i}$ for all $A\subseteq\{1,\dots,n\}$ and $i\notin A$, modulo the diamond relations

$$x_{A\cup\{i\},j}+x_{A,i}=x_{A\cup\{j\},i}+x_{A,j},\qquad x_{A\cup\{i\},j}\,x_{A,i}=x_{A\cup\{j\},i}\,x_{A,j}$$

for $i\neq j$ outside $A$) into the free skew field $\mathcal{C}=k\langle\langle x_1,\dots,x_n\rangle\rangle$, defined on generators by

$$e(x_{A,i})=V(x_{i_1},\dots,x_{i_k},x_i)\,x_i\,V(x_{i_1},\dots,x_{i_k},x_i)^{-1}\qquad(A=\{i_1,\dots,i_k\}),$$

with $V$ the Vandermonde quasideterminant, is injective for every $n\ge2$. The cases $n\le3$ are known, so the open content is every $n\ge4$.

Equivalent formulations of the same statement: (i) $Q_n\cong\hat Q_n$, where $\hat Q_n$ is the subalgebra of the free skew field generated by the pseudo-roots of a generic polynomial of degree $n$; (ii) every polynomial relation satisfied by the pseudo-roots $x_{A,i}$ in $\mathcal{C}$ is a consequence of the diamond relations; (iii) the kernel of $e$ is zero, that is, no nonzero element of $Q_n$ vanishes identically as a rational function of the generic roots $x_1,\dots,x_n$.

The Koszulity of $Q_n$, raised alongside this question in the source paper, is settled affirmatively and is not part of the present problem.

References

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