## Trace rings of generic matrices are Cohen-Macaulay.(English)Zbl 0697.20025

Let G be a reductive algebraic group over an algebraically closed field k of characteristic zero, and let W be a finite-dimensional representation of G. Let U be another finite-dimensional G-representation. A natural generalization of the Hochster-Roberts theorem would be that $$(U\otimes_ kk[W])^ G$$ is a Cohen-Macaulay $$k[W]^ G$$-module. Examples show that this cannot be true in general; however, a conjecture due to R. P. Stanley [Proc. Symp. Pure Math. 34, 345-355 (1979; Zbl 0411.22006)] gives at least some cases under which the above statement is true. The author replaces this conjecture with a slightly weaker one and proves this new conjecture for certain pairs (G,W) (namely for those where the locus of G-unstable points in W//G is constructible, i.e. it can be build up from smaller manageable parts in some sense, explained explicitly in the paper). It is proved that the conjecture holds if $$G=SL(2)$$.
If $$G=SL(V)$$, $$W=End(V)^{m*}$$, $$U=End(V)$$, then $$T_{m,n}=(U\otimes_ kk[W])^ G$$ is the noncommutative trace ring of m generic (n$$\times n)$$- matrices. The conjecture is satisfied in this case, which makes it possible for the author to prove that $$T_{m,n}$$ is Cohen-Macaulay.
Reviewer: V.L.Popov

### MSC:

 20G05 Representation theory for linear algebraic groups 16Rxx Rings with polynomial identity 16S50 Endomorphism rings; matrix rings 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 15A72 Vector and tensor algebra, theory of invariants

Zbl 0411.22006
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