Denominators for the Poincaré series of invariants of small matrices. (English) Zbl 0948.16018
Let \(F\) be a field of characteristic \(0\), and let \(R\) be the \(F\)-algebra generated by generic matrices \(X_1,\dots,X_m\) of size \(N\times N\), \(\overline C\) the algebra generated by the traces of elements of \(R\), and \(\overline R\) the algebra generated by \(R\) and \(\overline C\), regarding the members of \(\overline C\) as scalars. These algebras are multigraded and determine (symmetric) Poincaré series in variables \(t_1,\dots,t_m\). The paper studies the Poincaré series of \(\overline C\) and \(\overline R\), both of which are known to be rational functions. The authors determine explicit denominators for both series for \(N\leq 4\) and all \(m\), and for \(N\leq 3\) prove that these are the denominators of minimal degree.
Reviewer: Victor Petrogradsky (Ulyanovsk)
MSC:
| 16R30 | Trace rings and invariant theory (associative rings and algebras) |
Software:
SFReferences:
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