Hoeffding spaces and Specht modules
ESAIM: Probability and Statistics, Tome 15 (2011), pp. S58-S68

It is proved that each Hoeffding space associated with a random permutation (or, equivalently, with extractions without replacement from a finite population) carries an irreducible representation of the symmetric group, equivalent to a two-block Specht module.

DOI : 10.1051/ps/2010022
Classification : 05E10, 60C05
Keywords: exchangeability, finite population statistics, Hoeffding decompositions, irreducible representations, random permutations, Specht modules, symmetric group 
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     author = {Peccati, Giovanni and Pycke, Jean-Renaud},
     title = {Hoeffding spaces and {Specht} modules},
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     pages = {S58--S68},
     year = {2011},
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     url = {https://www.numdam.org/articles/10.1051/ps/2010022/}
}
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Peccati, Giovanni; Pycke, Jean-Renaud. Hoeffding spaces and Specht modules. ESAIM: Probability and Statistics, Tome 15 (2011), pp. S58-S68. doi: 10.1051/ps/2010022

[1] D.J. Aldous, Exchangeability and related topics. École d'été de Probabilités de Saint-Flour XIII. LNM 1117, Springer, New York (1983). | Zbl | MR

[2] M. Bloznelis, Orthogonal decomposition of symmetric functions defined on random permutations. Combin. Probab. Comput. 14 (2005) 249-268. | Zbl | MR

[3] M. Bloznelis and F. Götze, Orthogonal decomposition of finite population statistics and its applications to distributional asymptotics. Ann. Stat. 29 (2001) 353-365. | Zbl | MR

[4] M. Bloznelis and F. Götze, An Edgeworth expansion for finite population statistics. Ann. Probab. 30 (2002) 1238-1265. | Zbl | MR

[5] P. Diaconis, Group Representations in Probability and Statistics. IMS Lecture Notes - Monograph Series 11, Hayward, California (1988). | Zbl | MR

[6] J.J. Duistermaat and J.A.C. Kolk, Lie groups. Springer-Verlag, Berlin-Heidelberg-New York (1997). | Zbl | MR

[7] O. El-Dakkak and G. Peccati, Hoeffding decompositions and urn sequences. Ann. Probab. 36 (2008) 2280-2310. | Zbl | MR

[8] G.D. James, The representation theory of the symmetric groups. Lecture Notes in Math. 682, Springer-Verlag, Berlin-Heidelberg-New York (1978). | Zbl | MR

[9] G. Peccati, Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations. Ann. Probab. 32 (2004) 1796-1829. | Zbl | MR

[10] G. Peccati and J.-R. Pycke, Decompositions of stochastic processes based on irreducible group representations. Theory Probab. Appl. 54 (2010) 217-245. | Zbl | MR

[11] B.E. Sagan, The Symmetric Group. Representations, Combinatorial Algorithms and Symmetric Functions, 2nd edition. Springer, New York (2001). | Zbl | MR

[12] R.J. Serfling, Approximation Theorems of Mathematical Statistics. Wiley, New York (1980). | Zbl | MR

[13] J.-P. Serre, Linear representations of finite groups, Graduate Texts Math. 42, Springer, New York (1977). | Zbl | MR

[14] L. Zhao and X. Chen, Normal approximation for finite-population U-statistics. Acta Math. Appl. Sinica 6 (1990) 263-272. | Zbl | MR

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