Combinatorial $S\sb n$-modules as codes
Journal of Algebraic Combinatorics, Tome 4 (1995) no. 1, pp. 47-68.

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Summary: Certain $\mathbb Z S _{ n}$ mathbbZS_n -modules related to the kernels ofincidence maps between types in the poset defined by the natural productorder on the set of $n$-tuples with entries from 1, frac14 $\ldots , m$ are studied as linear codes (whencoefficients are extended to an arbitrary field $K$). Theirdimensions and minimal weights are computed. The Specht modules areextremal among these submodules. The minimum weight codewords of theSpecht module are shown to be scalar multiples of polytabloids. Ageneralization of t-design arising from the natural permutation $S _{n}$-modules labelled by partitions with $m$parts is introduced. A connection with Reed-Muller codes is noted and acharacteristic free formulation is presented.
Mots-clés : symmetric group, Specht module, $t$-design, Reed-muller code 
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     author = {Liebler, Robert A. and Zimmermann, Karl-Heinz},
     title = {Combinatorial $S\sb n$-modules as codes},
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     pages = {47--68},
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     year = {1995},
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Liebler, Robert A.; Zimmermann, Karl-Heinz. Combinatorial $S\sb n$-modules as codes. Journal of Algebraic Combinatorics, Tome 4 (1995) no. 1, pp. 47-68. https://geodesic-test.mathdoc.fr/item/JAC_1995__4_1_a1/