## 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.

### MSC:

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

### Citations:

Zbl 0411.22006; Zbl 0697.20025; Zbl 0703.15031
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