Matrices with polynomial or index-independent coefficients in the noncommutative Cayley–Hamilton theorem

For the generic $n\times n$ matrix $A=(a_{ij})$ over the free field, Gelfand–Krob–Lascoux–Leclerc–Retakh–Thibon (1995) proved a noncommutative Cayley–Hamilton theorem $\sum_j(-1)^jL_j(A)A^{n-j}=0$, where $L_j(A)$ is a diagonal matrix whose entries $L_j^{(i)}(A)$ are the coefficients of the $n$ pseudo-characteristic polynomials $Q_i(t)$ obtained from the rationality of the series $|I-tA|_{ii}^{-1}$. Generically these coefficients are complicated rational expressions involving inverses of the entries, and they depend on the row index $i$. The paper poses the problem of investigating two classes of specializations of the generic matrix: (1) matrices for which all $L_j^{(i)}(A)$ are polynomials in the entries, and (2) matrices for which $L_j^{(i)}(A)$ is independent of $i$, so that the characteristic polynomial is essentially scalar. The matrix $E_n$ of standard generators of $U(gl_n)$ lies in both classes (its pseudo-determinants equal the Capelli determinant), while the quantum matrix $A_q$ of $GL_q(n)$ lies in the first only. Three decades of later work have produced many further example families — quantum matrix algebras of orthogonal, symplectic and super type, reflection-equation algebras, Manin and $q$-Manin matrices — together with neighbouring universal-identity frameworks, but no necessary-and-sufficient characterization of either class is known. The problem of systematically characterizing these two classes remains open.

Background

The classical Cayley–Hamilton theorem states that a square matrix over a commutative ring is annihilated by its own characteristic polynomial. When the entries of a matrix do not commute, there is no single determinant and the theorem has no obvious analogue. A systematic noncommutative linear algebra was built from the quasi-determinant: for a matrix $A=(a_{ij})$ with noncommuting invertible entries, the $(i,j)$-quasi-determinant $|A|_{ij}$ is the inverse of the $(j,i)$-entry of $A^{-1}$, and it behaves as a noncommutative analogue of a ratio of two determinants (Gelfand–Retakh, Selecta Math. 3 (1997) 517–546). Rational expressions in noncommuting variables live naturally in the free field, the universal skew field of fractions of the free algebra $K\langle a_{ij}\rangle$, which plays the role of the field of rational functions.

Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon developed the theory of noncommutative symmetric functions — the free graded algebra $\operatorname{Sym}=K\langle\Lambda_1,\Lambda_2,\ldots\rangle$ on noncommuting elementary symmetric functions — and, in Section 8.6 of their 1995 paper, applied it to the characteristic polynomial of the generic matrix. For the generic $n\times n$ matrix $A=(a_{ij})$ (the entries are free generators over a field $K$ of characteristic zero) and a diagonal index $i$, set

$$\sigma(t,\alpha_i)=\sum_{k\ge0}S_k(\alpha_i)t^k=|I-tA|_{ii}^{-1},$$

where the complete symmetric functions $S_k(\alpha_i)$ of the formal alphabet $\alpha_i$ attached to the diagonal index $i$ are defined by this generating series. It is a noncommutative analogue of the ratio $\det(I-tA_{\hat i})/\det(I-tA)$, where $A_{\hat i}$ is obtained from $A$ by deleting row and column $i$. This series is rational over the free field: one has $\sigma(t,\alpha_i)=Q_i(t)^{-1}P_i(t)$ with $Q_i,P_i$ of degrees $n$ and $n-1$, normalized by $Q_i(0)=P_i(0)=1$. Writing

$$Q_i(t)=\sum_{0\le j\le n}(-1)^jL_j^{(i)}(A)\,t^j,$$

the coefficients $L_j^{(i)}(A)$ — expressible by Hankel-type quasi-determinants in the $S_k(\alpha_i)$ and also by a noncommutative Krylov formula in the entries of the powers of $A$ — satisfy the noncommutative Cayley–Hamilton theorem

$$\sum_{0\le j\le n}(-1)^jL_j(A)\,A^{n-j}=0,$$

where $L_j(A)=\operatorname{diag}(L_j^{(1)}(A),\ldots,L_j^{(n)}(A))$. The leading coefficients $\det_i(A)=L_n^{(i)}(A)$ are the $n$ pseudo-determinants of $A$, exact noncommutative analogues of the determinant. Already for $n=2$,

$$Q_1(t)=1-(a_{11}+a_{12}a_{22}a_{12}^{-1})t+(a_{12}a_{22}a_{12}^{-1}a_{11}-a_{12}a_{21})t^2,$$

so the coefficients are genuine rational expressions involving inverses of entries, and they depend on the index $i$: the characteristic polynomial has diagonal-matrix rather than scalar coefficients.

Two specializations behave much better. If $E_n=(e_{ij})$ is the matrix of standard generators of the universal enveloping algebra $U(gl_n)$ of the general linear Lie algebra (the associative algebra generated by the matrix units $e_{ij}$ subject to the commutation relations of $gl_n$), then all $Q_i(t)$ coincide with $\det(I-t(E_n+(n-1)I))$; their coefficients lie in the center of $U(gl_n)$, the pseudo-determinants reduce to the classical Capelli determinant (a central element of $U(gl_n)$ constructed by Capelli), and one recovers the characteristic identities satisfied by the generators on highest-weight modules (the standard class of $gl_n$-modules classified by a maximal weight). If $A_q=(a_{ij})$ is the matrix of generators of the quantum group $GL_q(n)$ (the $q$-deformed analogue of the algebra of polynomial functions on the general linear group), whose entries satisfy quadratic $q$-commutation relations, then the pseudo-determinants are all equal, up to powers of $q$, to the quantum determinant (the central determinant-like element of that algebra), every coefficient of every $Q_i(t)$ is a polynomial in the $a_{ij}$, and one recovers the quantum Cayley–Hamilton theorem; the lower coefficients still differ by powers of $q$ from one diagonal position to another. In view of these examples the authors pose the problem stated below.

Problem Statement

As posed at the end of Section 8.6 of the paper of Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon (1995), systematically characterize the following two classes of $n\times n$ matrices with noncommutative entries — that is, of specializations of the generic matrix, where the coefficients $L_j^{(i)}(A)$ of the pseudo-characteristic polynomials $Q_i(t)$ are computed in the skew field generated by the entries via the rationality of $\sigma(t,\alpha_i)=|I-tA|_{ii}^{-1}$:

1. Matrices $A$ for which every coefficient $L_j^{(i)}(A)$, $0\le j\le n$ and $1\le i\le n$, is a polynomial in the entries $a_{ij}$, so that no inverses of entries survive in the pseudo-characteristic polynomial.

2. Matrices $A$ for which the coefficients $L_j^{(i)}(A)$ do not depend on $i$, so that the coefficients of the noncommutative characteristic polynomial are scalars rather than diagonal matrices.

Characterizing the two classes means determining necessary and sufficient conditions for membership in each class. Investigating the classes includes clarifying how the two classes relate to one another and to the known example families (enveloping algebras, quantum groups, and Manin-type matrices, that is, matrices whose entries satisfy Manin's quadratic cross-commutation relations), and what obstruction makes the generic matrix fail both properties.

References

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