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The structure of spaces of Toeplitz matrices. (English) Zbl 0894.57016
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 [C. P. Boyer, J. C. Hurtubise, B. M. Mann and R. J. Milgram, Ann. Math., II. Ser. 137, No. 3, 561-609 (1993; Zbl 0816.55002)].

MSC:
57N65 Algebraic topology of manifolds
32G13 Complex-analytic moduli problems
55P35 Loop spaces
58D27 Moduli problems for differential geometric structures
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