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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

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