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Zbl 0930.16013
Aslaksen, Helmer; Tan, Eng-Chye; Zhu, Chen-bo
Invariant theory of matrices. (English)
[A] Gruber, Bruno (ed.), Symmetries in science VIII. Proceedings of a symposium, Bregenz, Austria, August 1994. New York, NY: Plenum Press. 13-19 (1995). ISBN 0-306-45119-0

Let $F$ be a field of characteristic $0$, let $M(n,m)=M(n,m,F)$ denote the set of $n\times m$ matrices over $F$ and let $W=W(n,m,F)$ be the vector space of $m$-tuples of $n\times n$ matrices over $F$. Let $V\subset W$ be a vector space on which a group $G\subset\text{GL}(n,F)$ acts by simultaneous conjugation. We will denote the polynomial functions on $V$ by $P(V)$ and the $G$ invariants by $P(V)^G$.\par Main theorem: When $G=\text{SO}(2k+1,F)$, the invariants $P[W(2k+1,m,F)]^G$ are the same as the $O(2k+1,F)$ invariants. When $G=\text{SO}(2k,F)$, the invariants $P[W(2k,m,F)]^G$ are generated by traces and polarized Pfaffians of the $A_i$ and $A^t_i$, $i=1,\dots,m$, i.e., $\text{tr }P(A,A^t)$ and $\text{pl}(P_1(A,A^t),\dots,P_k(A,A^t))$, where $P,P_1,\dots,P_k$ are noncommutative polynomials and $A\in W(2k,m,F)$.

MSC 2000:: *16R30 Trace rings and invariant theory (assoc. rings and algebras)
16U80 Generalizations of commutativity (assoc. rings and algebras)
16U70 Commutativity theorems for assoc. rings

Keywords: tuples of matrices; polynomial functions; invariants; traces; polarized Pfaffians

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