Group actions on Stanley-Reisner rings and invariants of permutation groups.

*(English)*Zbl 0561.06002This 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.

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 |

##### Keywords:

symmetric group; basic sets; free basis; symmetric polynomials; invariant polynomials; Stanley-Reisner rings; Coxeter complexes; ranked poset; chain complex; order preserving automorphisms; elementary symmetric functions; balanced complex; poset rings; finite Coxeter groups; shellability; parabolic subgroup; Weyl groups; dominant weights
PDF
BibTeX
XML
Cite

\textit{A. M. Garsia} and \textit{D. Stanton}, Adv. Math. 51, 107--201 (1984; Zbl 0561.06002)

Full Text:
DOI

##### References:

[1] | Baclawski, K, Rings with lexicographic straightening law, Advan. in math., 39, 185-213, (1981) · Zbl 0466.13004 |

[2] | Baclawski, K; Garsia, A.M, Combinatorial decompositions of a class of rings, Advan. in math., 39, 155-184, (1981) · Zbl 0466.13003 |

[3] | Bjorner, A, Shellable and Cohen-Macaulay partially ordered sets, Trans. amer. math. soc. 2, 60, 1, 159-183, (1980) · Zbl 0441.06002 |

[4] | \scA. Bjorner, Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, U. of Stockholm, preprint. · Zbl 0546.06001 |

[5] | Bjorner, A; Wachs, M, Bruhat order of Coxeter groups and shellability, Advan. in math., 43, 87-100, (1982) · Zbl 0481.06002 |

[6] | \scA. Bjorner and M. Wachs, On lexicographically shellable posets, preprint. · Zbl 0514.05009 |

[7] | Bourbaki, N, Groupes et algèbres de Lie, (), Chaps. 4-6 · Zbl 0483.22001 |

[8] | Brugesser, H; Mani, P, Shellable decompositions of cells and spheres, Math. scand., 29, 197-205, (1971) · Zbl 0251.52013 |

[9] | Carter, R.W, Simple groups of Lie type, (1972), Wiley New York · Zbl 0248.20015 |

[10] | Danaraj, G; Klee, V, Which spheres are shellable?, Ann. discrete math., 2, 33-52, (1978) · Zbl 0401.57031 |

[11] | DeConcini, C; Procesi, C, Hodge algebras: A survey, () |

[12] | Garsia, A.M; Gessel, I, Permutation statistics and partitions, Advan. in math., 31, 3, 258-305, (1979) · Zbl 0431.05007 |

[13] | Garsia, A.M, Methodes combinatoires dans la théorie des anneaux de Cohen-Macaulay, C. R. acad. sci. Paris ser. A, 288, 371-374, (1979) · Zbl 0401.13014 |

[14] | Garsia, A.M, Combinatorial methods in the theory of Cohen-Macaulay rings, Advan. in math., 38, 229-266, (1980) · Zbl 0461.06002 |

[15] | Garst, P.F, Cohen-Macaulay complexes and group actions, () |

[16] | Humphreys, J.E, Introduction to Lie algebras and representation theory, (1972), Springer-Verlag New York · Zbl 0254.17004 |

[17] | Reisner, G, Cohen-Macaulay quotients of polynomial rings, Advan. in math., 21, 30-49, (1976) · Zbl 0345.13017 |

[18] | Sloane, N.J.A, Error-correcting codes and invariant theory: new applications of a nineteenth-century technique, Amer. math. monthly, 84, 82-107, (1977) · Zbl 0357.94014 |

[19] | Solomon, L, A decomposition of the group algebra of a finite Coxeter group, J. algebra, 9, 220-239, (1968) · Zbl 0186.04503 |

[20] | Solomon, L, The Steinberg character of a finite group with a BN-pair, (), 213-221 · Zbl 0216.08001 |

[21] | Stanley, R, Cohen-Macaulay complexes, (), 51-52 |

[22] | Stanley, R, Relative invariants of finite groups generated by pseudo-reflections, J. algebra, 49, 1, 134-148, (1977) · Zbl 0383.20029 |

[23] | Stanley, R, Hilbert functions of graded algebras, Advan. in math., 28, 57-83, (1978) · Zbl 0384.13012 |

[24] | Stanley, R, Balanced Cohen-Macaulay complexes, Trans. amer. math. soc., 249, 139-157, (1979) · Zbl 0411.05012 |

[25] | Stanley, R, Invariants of finite groups and their applications to combinatorics, Bull. amer. math. soc., 1, 475-511, (1979), (N. S.) · Zbl 0497.20002 |

[26] | \scR. Stanley, Some aspects of groups acting on finite posets, preprint. · Zbl 0496.06001 |

[27] | \scR. Stanley, Combinatorics and invariant theory, preprint. |

[28] | Steinberg, R, Lectures on Chevalley groups, () · Zbl 1361.20003 |

[29] | Steinberg, R, On a theorem of pittie, Topology, 14, 173-177, (1975) · Zbl 0318.22010 |

[30] | \scM. Wachs, Quotients of Coxeter complexes and buildings with linear diagram, preprint. · Zbl 0616.51009 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.