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Group actions on Stanley-Reisner rings and invariants of permutation groups. (English) Zbl 0561.06002
This paper is an interesting and important addition to the recent literature on the relationship of certain combinatorial structures with ring and group theory. It will be difficult to do justice to its contents in a short review.
Let \(H\) be a subgroup of the symmetric group \(S_ n\), then the main purpose of this paper is to give some combinatorial methods for the explicit construction of basic sets, that is, a free basis as a module over the symmetric polynomials, for the ring \({\mathbb{Q}}^ H[x_ 1,...,x_ n]\) of \(H\)-invariant polynomials in \(x_ 1,...,x_ n\) with rational coefficients. The authors say that their work was stimulated by the well known review papers by N. J. A. Sloane [Am. Math. Mon. 84, 82-107 (1977; Zbl 0357.94014)], R. Stanley [Bull. Am. Math. Soc., New. Ser. 1, 475-511 (1979; Zbl 0497.20002)], a paper by A. Björner where the combinatorial study of Stanley-Reisner rings of Coxeter complexes was established [Adv. Math. 52, 173-212 (1984; Zbl 0546.06001)] and R. Steinberg who had pointed out that earlier work of his [Topology 14, 173-177 (1975; Zbl 0318.22010)] was connected with an earlier paper of the first author [Adv. Math. 38, 229-266 (1980; Zbl 0461.06002)]. Their aim is to find a common setting for all the basic sets given by Garsia and Steinberg. The work develops that initiated by Garsia on the use of Stanley-Reisner rings; in fact, it is shown that an extension of the construction given by Garsia, combined with the shellability results of Björner, gives a natural common setting for the basic sets obtained earlier by Garsia and Steinberg. The details of the work are best described by the authors in their introduction.
“Initially our ingredients are a ranked poset \(P\), its chain complex \(C(P)\), the corresponding Stanley-Reisner ring \(R_ P\), and a group \(G\) of rank and order preserving automorphisms of \(P\). Our first goal is to study, for each subgroup \(H\subseteq G\), the subring \(R^ H_ P\) consisting of all \(H\)-invariant elements of \(R_ P\).
If \(P\) has \(d\) ranks, for each subset \(S\subseteq [d]\) we consider the rank selected subcomplex \(C_{=S}(P)\) consisting of all chains of \(P\) which hit everyone of the ranks \(i\in S\). If \(G\) acts transitively on \(C_{=S}(P)\) for every \(S\subseteq [d]\) then \(R^ G_ P\) has a very simple structure. Indeed, denoting by \(\Theta_ i\) the sum of the elements of the ith rank row of \(P\), we can show that each element of \(R^ G_ P\) is a polynomial in \(\Theta_ 1,\Theta_ 2,...,\Theta_ d\). It will be seen in the sequel that \(\Theta_ 1,\Theta_ 2,...,\Theta_ d\) play the same role as the elementary symmetric functions. More precisely, if \(P\) is Cohen-Macaulay, we can show that for any \(H\subseteq G\) we can construct polynomials \(\Delta_ 1,\Delta_ 2,...,\Delta_ N\) such that every element \(P\in R^ H_ P\) can be uniquely expressed in the form \(P=\sum^{N}_{i=1}\Delta_ ip_ i(\Theta_ 1,...,\Theta_ d)\). We refer to \(\{\Delta_ 1,...,\Delta_ N\}\) as a basic set for \(R^ H_ P\).
This given, the case when \(P\) is the \(n\)-subset lattice \(B_ n\) and \(G\) is the symmetric group \(S_ n\) is of special interest to us. Indeed, our program for constructing a basic set for a ring \(Q^ H[x_ 1,...,x_ n]\) is to construct one for \(R^ H_{B_ n}\) and then transfer it from \(R_{B_ n}\) to \(Q[x_ 1,...,x_ n]\). To increase the scope of the theory, in the latter part of our presentation we drop the requirement of an underlying poset, and work with a balanced complex \(C\). Doing so presents only minor changes since the Stanley-Reisner ring \(R_ C\) of such a complex behaves very much like a poset ring \(R_ P\). In this setting the most remarkable structures are the Coxeter complexes. Crudely speaking, we can work with any finite Coxeter group \(W\) and its corresponding Coxeter complex \(C(W)\) just as well as with the pair consisting of \(S_ n\) and the chain complex \(C(B_ n)\).
It develops that this is precisely the unifying setting we are looking for. This comes about as follows. By extending Björner’s shellability results, we obtain first a general construction which yields basic sets for the rings \(R^ H_{C(W)}\) when \(H\) is a parabolic subgroup. Then, when \(W\) is a Weyl group, and \(\lambda_ 1,\lambda_ 2,...,\lambda_ n\) is a fundamental system of dominant weights, by formally setting \(e^{\lambda_ i}=z_ i\), we can define an action of \(W\) on the ring \[ R'=Q[z_ 1,...,z_ n;1/z_ 1,...,1/z_ n]. \] On the other hand the system \(\lambda_ 1,...,\lambda_ n\) and its images under \(W\) may be used to give a concrete representation of the Coxeter complex \(C(W)\). From this circumstance we derive that the ring \(R_{C(W)}\) dominates \(R'\). This given, we can transfer a basic set for the ring \(R^ H_{C(W)}\) into a basic set for \(R'{}^ H\). It turns out that applying this transfer to the basic sets obtained by shellability methods yields precisely the basic sets of Steinberg.”
The paper concludes with an application to the representation of Coxeter groups and in particular, a solution to a problem posed by R. Stanley is answered.
Reviewer: A.O.Morris

MSC:
06A06 Partial orders, general
20C30 Representations of finite symmetric groups
15A72 Vector and tensor algebra, theory of invariants
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
05A19 Combinatorial identities, bijective combinatorics
55U15 Chain complexes in algebraic topology
55U10 Simplicial sets and complexes in algebraic topology
57Q05 General topology of complexes
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