ABSTRACT
In this chapter, we present the foundations of the so-called dual combinatorics of the characters of the symmetric groups. This theory relies on the following fundamental idea. In the first part of this book (in particular, Chapters 2 and 3), we detailed the representation theory of the 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/eq4711.tif"/> , and we constructed the characters ch λ , which are functions from 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/eq4712.tif"/> to ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq4713.tif"/> , or more precisely, from the set Y ( n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq4714.tif"/> of conjugacy classes 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/eq4715.tif"/> to ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq4716.tif"/> (and in fact to ℤ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq4717.tif"/> ): ch λ : μ ∈ Y ( n ) ↦ ch λ ( μ ) ∈ ℂ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371016/5e644d52-ef68-4f0e-ac73-bbf480af375a/content/eq4718.tif"/>
