Combinatorial S n - Modules as Codes *

Certain ZSn -modules related to the kernels of incidence maps between types in the poset defined by the natural product order on the set of n-tuples with entries from {1,. . . , m} are studied as linear codes (when coefficients are extended to an arbitrary field K). Their dimensions and minimal weights are computed. The Specht modules are extremal among these submodules. The minimum weight codewords of the Specht module are shown to be scalar multiples of poly tabloids. A generalization of t-design arising from the natural permutation Sn-modules labelled by partitions with m parts is introduced. A connection with Reed-Muller codes is noted and a characteristic free formulation is presented. This paper has two purposes, the second of which grew naturally out of the first. Our first purpose is to fill a number of gaps in [11] (cf. Remarks 2.7, 2.13). The second purpose is to point out and extend certain connections between combinatorial t-designs, representations of the symmetric group Sn labelled by 2-part partitions, and the classical Reed-Muller codes. We consider the natural permutation representations of the symmetric group to be afforded by Z-free modules over ZSn and we work with Z-pure submodules over this ring whenever possible. This approach differs from that of many authors who take the coefficient ring to be an arbitrary field K. Our approach has many technical advantages and results about KSn-modules can be obtained from ours by "extending coefficients" (tensoring with K over Z). The basis of this work is the poset of flags introduced in [11]. Some of its elementary properties as well as a number of alternative formulations are presented in Section 2.1. The incidence maps of this poset are used to introduce Z-forms on certain classical QSnmodules in Section 2.2, and our principal object of study ZBkL is described in Theorem 2.4 and Corollary 2.5. The poset is used to construct large collections of parity checks defined over Z in Section 2.3. Section 3 presents the most important special cases of the results of Section 4 and may serve as an introduction to this section. The final section presents the combinatorial and coding theoretic connections mentioned above. This research was partially supported by NSA grant 904-91-H-0048.