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The normality of closures of conjugacy classes of matrices. (English) Zbl 0822.20045

Let \(K\) be an algebraically closed field, \(n\) a positive integer, \(x\) an \(n \times n\) matrix with entries in \(K\), let \(C(x)\) be the orbit of \(x\) under the adjoint action of \(\text{GL} (n,K)\) and let \(\overline {C(x)}\) be the Zariski closure of \(C(x)\). H. Kraft and C. Procesi [ibid. 53, 227-247 (1979; Zbl 0434.14026)], have proved that if \(K\) has characteristic 0 then \(\overline {C(x)}\) is a normal, Cohen-Macaulay variety with rational singularities. The main purpose of this paper is to prove, in arbitrary characteristic, that \(\overline {C(x)}\) is normal.

MSC:

20G05 Representation theory for linear algebraic groups
14L30 Group actions on varieties or schemes (quotients)
20G15 Linear algebraic groups over arbitrary fields
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)

Citations:

Zbl 0434.14026
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References:

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