ABSTRACT
In the third part of the book, we developed a theory of observables of Young diagrams, which allowed us to make computations with characters of symmetric groups that were somewhat generic, in the sense that the dependence in n of these computations was well understood. We now want to use this theory in order to find the typical properties of large representations of symmetric groups. To make this program more precise, consider a symmetric group S ( n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq6837.tif"/> with n large, and a “natural” representation (V, ρV ) of S ( n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq6838.tif"/> , e.g., the regular representation of S ( n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq6839.tif"/> on ℂ S ( n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq6840.tif"/> , or the representation by permutation of the letters of words in the tensor product ( ℂ N ) ⊗ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq6841.tif"/> . This representation V is in general reducible, and its normalized trace χ V ( ⋅ ) = tr ρ V ( ⋅ ) dim V https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq6842.tif"/> decomposes as a barycenter of the irreducible characters χ V = ∑ λ ∈ Y ( n ) ℙ V [ λ ] χ λ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq6843.tif"/>
