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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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