A converse to Stanley’s conjecture for \(Sl_ 2\). (English) Zbl 0817.13004

Let \(G = Sl(V)\) where \(V\) is a two-dimensional vector space over an algebraically closed field \(k\) of characteristic zero. Define \(W = \bigoplus^ m_{i=1} S^{d_ i}V\), \(d = \dim W = \sum (d_ i + 1)\), and \(R = SW\), where \(SW\) denotes the symmetric algebra of \(W\). Define for \(n \geq 0\), \(s^{(n)} = n + (n-2) + \cdots + 1 = {(n + 1)^ 2 \over 4}\) if \(n\) is odd, \(s^{(n)} = n + (n-2)+\cdots+2={n(n+2)\over 4}\) if \(n\) is even, and put \(s = \sum^ m_{i=1} s^{(d_ i) }\). It follows from a conjecture of R. P. Stanley [Proc. Symp. Pure Math. Am. Math. Soc., Columbus 1978, Proc. Symp. Pure Math. 34, 345-355 (1979; Zbl 0411.22006)] that \((R \otimes S^ \mu V)^ G\) is Cohen-Macaulay if \(\mu < s - 2\). This conjecture was proved in almost complete generality by the author [J. Am. Math. Soc. 2, No. 4, 775-799 (1989; Zbl 0697.20025)].
B. Broer proved [Indag. Math., New Ser. 1, No. 1, 15-25 (1990; Zbl 0703.15031)] a partial converse to Stanley’s conjecture for \(Sl_ 2\). In this note we will prove a complete converse.


13C14 Cohen-Macaulay modules
14L24 Geometric invariant theory
13A50 Actions of groups on commutative rings; invariant theory
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