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The Hilbert series of matrix concomitants and its application. (English) Zbl 0759.16011
Perspectives in ring theory, Proc. NATO Adv. Res. Workshop, Antwerp/Belg. 1987, NATO ASI Ser., Ser. C 233, 31-36 (1988).
[For the entire collection see Zbl 0676.00006.]
Let $$G$$ be a semisimple algebraic group over the complex numbers $$\mathbb{C}$$, let $$\mathfrak G$$ be the Lie algebra of $$G$$, and let $$W_ m$$ be the direct sum of $$m$$ copies of $$\mathfrak G$$, with the adjoint action of $$G$$. The fixed ring of the action of $$G$$ on the symmetric algebra $$\mathbb{C}[W_ m]$$ is denoted $$C(G,m)$$. The author shows that the Hilbert series of $$C(G,m)$$ satisfies a functional equation (Theorem 1).
The trace ring of $$m$$ generic $$n\times n$$ matrices, $$T(n,m)$$, is a noncommutative ring which is a finite module over $$C(GL(n),m)$$. He shows that the Hilbert series of $$T(n,m)$$ satisfies a functional equation (Theorem 4) and uses this fact to re-prove the following theorem of L. Le Bruyn and C. Procesi [Lect. Notes Math. 1271, 143-175 (1987; Zbl 0634.14034)]. The global dimension of $$T(n,m)$$ is infinite except in the following cases: $$n=1$$, $$m=1$$, $$(n,m)=(2,2),\;(2,3)$$ or $$(3,2)$$.
##### MSC:
 16R30 Trace rings and invariant theory (associative rings and algebras) 15A72 Vector and tensor algebra, theory of invariants