an:01018424
Zbl 0894.57016
Milgram, R. J.
The structure of spaces of Toeplitz matrices
EN
Topology 36, No. 5, 1155-1192 (1997).
00039976
1997
j
57N65 32G13 55P35 58D27
space of Toeplitz matrices; moduli space; instantons; stratification; Toeplitz matrix
A finite Toeplitz matrix is a matrix \(t_n = (a_{i,j})\) with the coefficients of the form \(a_{i,j} = a_{i-j}\). Consider the variety of projective equivalence classes of nonzero \(n\times n\)-Toeplitz matrices \((t_n\equiv \alpha t_n\) for all nonzero \(\alpha\in\mathbb{C}\)), which are parameterized by the complex projective space \(\mathbb{C} P^{2n-2}\). The paper is devoted to a study of subvarieties \(T_{n,k}\) consisting of those \(t_n\) with \(\dim(\text{kernel}(t_n))\geq k\), and the open varieties \({\mathcal T}_{n,k} = T_{n,k}-T_{n,k+1}\). The results are summarized in the following three theorems.
Theorem A. The space \({\mathcal T}_{n,0}\) of nonsingular \(n\times n\)-Toeplitz matrices is homeomorphic to the orbit space under the action of \(GL_2(\mathbb{C})\) on the space of pairs \((p_1(z),p_2(z))\) of coprime polynomials with \(\max(\text{deg}(p_1(z)),\text{deg}(p_2(z)))=n\), where the action is given as
\[
(p_1\;p_2) {{a\;c}\choose {b\;d}}= (ap_1+ bp_2\;cp_1+ dp_2).
\]
Theorem B. Let \(t_n\in {\mathcal T}_{n,k}\). Then there is a vector \(w\in\text{Ker}(t_n)\) so that
\[
\text{Ker}(t_n)=\langle w,s(w),\dots, s^{k -1}(w)\rangle,
\]
where \(s\) is a shift operator. Moreover, (1) the first \(k-1\) coordinates of \(w\) are zero; (2) \(w\) is unique up to multiplication by a nonzero scalar.
Theorem C.
\[
\begin{aligned} H^*({\mathcal T}_{n,k};\mathbb{Q})& \cong H^*(S^2;\mathbb{Q}), k\geq 1\\ H^*({\mathcal T}_{n,0};\mathbb{Q})& \cong H^*(pt;\mathbb{Q}).\end{aligned}
\]
The rational cohomology of the unprojectivized versions of \({\mathcal T}_{n,k}\) are also computed. These results lead to a complete determination of the rational cohomology of the strata in a stratification of the moduli spaces \({\mathcal M}_k\) of gauge equivalence classes of SU(2)-Yang-Mills instantons on \(S^4\). This stratification was described in [\textit{C. P. Boyer, J. C. Hurtubise, B. M. Mann} and \textit{R. J. Milgram}, Ann. Math., II. Ser. 137, No. 3, 561-609 (1993; Zbl 0816.55002)].
I.Itenberg (Rennes)
Zbl 0816.55002