The Hilbert series of matrix concomitants and its application. (English) Zbl 0759.16011

Teranishi, Yasuo

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

Keywords:

semisimple algebraic group; adjoint action; fixed ring; symmetric algebra; Hilbert series; functional equation; trace ring; generic \(n\times n\) matrices; global dimension