Unspecified Journal Volume 00, Number 0, Pages 000-000 S ????-????(XX)0000-0 DEGREE BOUNDS AND THE REGULAR REPRESENTATION: APPENDIX MARA D. NEUSEL Abstract.Let G be a matrix group consisting of permutation matrices. Let F and K be two different fields. We show that if the polynomial invarian* *ts F[V ]G and K[V ]G are both Cohen-Macaulay, then they are simultaneously Gorenstein, complete intersections, hypersurfaces, resp. polynomial. T* *hus Cohen-Macaulay rings of permutation invariants are polynomial exactly wh* *en G is generated by pseudo-reflections. 1.Appendix Let G be a matrix group consisting of n x n-permutation matrices. As such we can consider G as a subgroup of GL(n, F) for any field F. Denote by F and K two different fields. In [2] it has been shown that if the polynomial invariants F[V ]G and K[V ]G are both Cohen-Macaulay, then there exi* *sts a common minimal algebra generating set (consisting of orbit sums of monomials) for both rings, see Theorem 2.8 in ibid. In this appendix we want to prove some more immediate corollaries from that. In particular, in this appendix we are interested in the fine classification of* * Cohen- Macaulay rings. Proposition 1.1. If F[V ]G and K[V ]G are both Cohen-Macaulay, then they are simultaneously Gorenstein. Proof.This follows immediately from a result due to Stanley: If F[V ]G is Cohen- Macaulay, then it is Gorenstein if and only if its Poincar'e series satisfies t* *he following condition P (F[V ]G , 1_t) = (-1)ntsP (F[V ]G , t) for some integer s, see Theorem 5.7.5 in [3]. Since the Poincar'e series is ind* *ependent of the ground field, see Proposition 3.2.2 in [3], the result follows. Next we turn to complete intersections. Proposition 1.2. If F[V ]G and K[V ]G are Cohen-Macaulay, then they are simul- tenaously complete intersections. ____________ Received by the editors August 17, 2007. 2000 Mathematics Subject Classification. Primary 13A50, Secondary. Key words and phrases. Invariant Theory of Finite Permutation Groups, Permut* *ation Repre- sentation, Cohen-Macaulay, Gorenstein, Complete Intersection, Hypersurface, Pol* *ynomial Alge- bra, Pseudo-Reflection. Oc0000 (copyright holder) 1 2 MARA D. NEUSEL Proof.As in the result above we use the fact that the Poincar'e series of invar* *iant rings of permutations groups is independent of the ground field. Let {f1, . .,.fk} be a common minimal algebra generating set for F[V ]G and K[V ]G . Let F[V ]G be a complete intersection. Then F[V ]G = F[f1, . .,.fk]=(r1, . .,.rk-n), where r1, . .,.rk-n is a regular sequence. Thus |r1|) . .(.1 - t|rk-n|) P (F[V ]G , t) = (1_-_t_______________ = P (K[V ]G , t), (1 - t|f1|) . .(.1 - t|fk|) where | - | denotes the degree of the polynomial -, see Lemma 2.2 in [1]. If both fields have the same characteristic there is nothing to show. Assume that F has characteristic zero and K has finite characteristic p. A re* *lation riamong the algebra generators f1, . .,.fk valid in characteristic zero remains* * valid in finite characteristic. Thus the remembering map OE : K[F1, . .,.Fk]=(R1, . .,.Rk-n) -! K[V ]G , OE(Fi) = fi i = 1, . .,.k, is surjective. Therefore the Poincar'e series of the algebra on the left must b* *e term- wise greater or equal to the Poincar'e series of the algebra on the right: P (K[F1, . .,.Fk]=(R1, . .,.Rk-n), t) >> P (K[V ]G , t). However, by construction we have P (F[V ]G , t) = P (K[F1, . .,.Fk]=(R1, . .,.Rk-n), t) >> P (K[V ]G , t) = P (F* *[V ]G , t). Thus the Poincar'e series are equal and OE is an isomorphism. Finally assume that F has finite characteristic p and K has characteristic ze* *ro. The argument is similar to the preceding case: If r is a relation among the fi's modulo p. Then we lift it to characteristic zero and obtain r(f1, . .,.fk) = ph(f1, . .,.fk) for some invariant h 2 K[V ]G . Thus each rj, j = 1, . .,.k-n leads to a charac* *teristic zero equation rj- phj = 0 for suitable invariants hj. Thus we obtain a surjective remembering map _ : K[F1, . .,.Fk]=(Rj- pHj, j = 1, . .,.k - n) -! K[V ]G _(Fi) = fi i = 1, .* * .k.. As above we can conclude that the Poincar'e series of both rings are equal and * *_ is an isomorphism. Corollary 1.3. If F[V ]G and K[V ]G are Cohen-Macaulay, then they are simulta- neously hypersurfaces. Proof.This is the preceding result specialized to k = n + 1. Proposition 1.4. If F[V ]G and K[V ]G are Cohen-Macaulay, then they are simul- taneously polynomial algebras. Proof.If F[V ]G is polynomial then any minimal algebra generating set consists * *of exactly n polynomials, hence K[V ]G is polynomial. Since rings of invariants in characteristic zero are polynomial exactly when * *the group G is generated by pseudo reflections we have the following result. SCHMID'S RESULT 3 Corollary 1.5. Let G GL(n, F) be a permutation group. Assume that its ring of invariants, F[V ]G , is Cohen-Macaulay. Then F[V ]G is polynomial if and only i* *f G is generated by transpositions. References 1.M. D. Neusel, Invariants of some Abelian p-Groups in Characteristic p, Proce* *edings of the AMS 125 (1997), 1921-1931. 2.M. D. Neusel, Degree Bounds and the Regular Representation, preprint 2007. 3.Mara D. Neusel and Larry Smith, Invariant Theory of Finite Groups, Mathemati* *cal Surveys and Monographs Vol.94, AMS, Providence RI 2002. Department of Mathematics and Statistics, MS 1042, Texas Tech University, Lub- bock, Texas 79409 E-mail address: Mara.D.Neusel@ttu.edu