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Etale local strucure of matrix invariants and concomitants. (English) Zbl 0634.14034
Algebraic groups, Proc. Symp., Utrecht/Neth. 1986, Lect. Notes Math. 1271, 143-175 (1987).
[For the entire collection see Zbl 0619.00008.]
The authors study the variety \(V_{mn}\) built as an “approximation” of the space of orbits of \(X_{mn}=\oplus^{m}_{i=1}M_ n({\mathbb{C}})\quad under\) action by componentwise conjugation of \(GL_ n({\mathbb{C}})\) by using \(GL_ n({\mathbb{C}})\)-invariants as parameters of \(V_{mn}\). It parametrizes naturally the closed orbits of this action. The coordinate ring of \(V_{mn}\) is the so-called trace ring \(\pi_{mn}\) of m generic \(n\times n\)-matrices [see C. Procesi; Adv. Math. 19, 306-381 (1976; Zbl 0331.15021)]. A point \(\xi\) of \(V_{mn}\) is of representation type \(\tau =(e_ 1,k_ 1;...;e_ r,k_ r)\) provided the corresponding isomorphism class of semisimple representations is built from r distinct simple components of dimensions \(k_ i\) occuring with multipicities \(e_ i\). It is shown that \(V_{mn}(\tau)\), the subset of \(V_{mn}\) consisting of all points of representation type \(\tau\), form a finite stratification into locally closed smooth subvarieties, where \(V_{mn}(\tau)\) lies in the closure of \(V_{mn}(\tau ')\) if and only if \(\tau\) is a degeneration of \(\tau '\). Furthermore, there is an explicit determination of the étale local structure of points of representation type with multiplicities 1. It is shown that this yields a Cohen-Macaulay module and its Poincaré series satisfies a certain functional equation.
These results are used in order to show that the singular locus of \(V_{mn}\) is determined by the Formanek center of \(\pi_{mn}\). It concludes with the solution of the regularity problem for trace rings of generic matrices, i.e., \(gl\dim (\pi_{mn})<\infty\) if and only if m or n is equal to one or \((m,n)=(2,2), (2,3)\) or (3,2). The authors use the work of D. Luna [Bull. Soc. Math. Fr., Suppl., Mém. No.33, 81-105 (1973; Zbl 0286.14014)] and R. P. Stanley [Invent. Math. 68, 175- 193 (1982; Zbl 0516.10009)].
Reviewer: P.Schenzel

14M12 Determinantal varieties
20G05 Representation theory for linear algebraic groups
15A24 Matrix equations and identities
14L30 Group actions on varieties or schemes (quotients)