Document Zbl 1041.13006 - zbMATH Open

Let \(\rho : G \to GL(n,\mathbb F)\) be a representation of a finite group \(G\) over a field \(\mathbb F.\) The group \(G\) acts on \(V = \mathbb F^n\) and therefore also on \(\mathbb F[V],\) the algebra of polynomial functions. The algebra of coinvariants \(\mathbb F[V]_G\) is defined as the quotient of \(\mathbb F[V]\) by the ideal generated by all homogeneous \(G\)-invariant forms of strictly positive degree. Assume that the characteristic of \(\mathbb F\) is zero. By a result of Steinberg [see R. Kane, Can. Math. Bull. 37, No. 1, 82–88 (1994; Zbl 0805.51010)], it follows that \(\mathbb F[V]_G\) is a Poincaré duality algebra (i.e. a zero-dimensional graded Gorenstein \(\mathbb F\)-algbra) if and only if \(G\) is a pseudoreflection group. The main aim of the present paper is to explore the situation over fields of non-zero characteristic. There is an analogue to Steinberg’s result for \(n=2,\) a partial result for \(n=3,\) and a counterexample in the modular case for \(n=4.\) The counterexample is given by \(G = \mathbb Z/2 \mathbb Z\) acting on \(\mathbb F_2[x_1,x_2,y_1,y_2]\) by the generator via \(x_i \mapsto y_i, i = 1,2.\)

MSC:

13A50	Actions of groups on commutative rings; invariant theory
20F55	Reflection and Coxeter groups (group-theoretic aspects)

References:

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