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Invariants of several matrices. (English) Zbl 0826.20036

Let \(K\) be an algebraically closed field of arbitrary characteristic, \(n\) and \(m\) positive integers and let \(M(n)\) be the set of \(n \times n\)- matrices with entries in \(K\). The purpose of this paper is to give generators for the algebra of polynomial invariants on the set of \(m\)- tuples of matrices \(M(n)^m = M(n) \times M(n) \times \cdots \times M(n)\) (\(m\)-times) for the action, by simultaneous conjugation, of the general linear group \(\text{GL} (n)\). Indeed we shall show that the polynomial invariants are generated by the functions \((x_1, x_2, \dots, x_m) \mapsto \text{Trace} (x_{i_1} x_{i_2} \dots x_{i_r}\), \(\bigwedge^s(E))\), for \((i_1, i_2, \dots, i_r)\) an \(r\)-tuple of integers in \(\{1,2,\dots, m\}\) (for some \(r\)), \(s\) a positive integer and \(E\) the natural \(M(n)\)-module. We shall derive the result from a more general one giving a description of the \(H\) class functions of \(G\) for a “saturated” subgroup \(H\) of a reductive group \(G\). This description involves certain shifted trace functions defined by partial tilting modules, which have been recently introduced into the theory by Ringel. In applying this general result to the situation above we take \(G\) to be \(\text{GL} (n)^m = \text{GL} (n) \times \text{GL} (n) \times \cdots \times \text{GL} (n)\) (\(m\) times) and \(H\) the diagonal embedding of \(\text{GL} (n)\). The partial tilting modules are then \(G\)- module summands of tensor products of exterior powers of the natural module and the module affording the dual of the determinant representation. The shifted trace functions are then shown to belong to the ring generated by the trace functions on exterior powers described above by means of a certain identity similar to one occurring in the theory of symmetric functions.

MSC:

20G05 Representation theory for linear algebraic groups
05E10 Combinatorial aspects of representation theory
19A22 Frobenius induction, Burnside and representation rings
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
14L24 Geometric invariant theory
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References:

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