We present a simple, explicit orthogonal basis of eigenvectors for the Johnson and Kneser graphs, based on Young's orthogonal representation of the symmetric group. Our basis can also be viewed as an orthogonal basis for the vector space of all functions over a slice of the Boolean hypercube (a set of the form ${(x_1,ldots,x_n) in {0,1}^n : sum_i x_i = k}$), which refines the eigenspaces of the Johnson association scheme; our basis is orthogonal with respect to any exchangeable measure. More concretely, our basis is an orthogonal basis for all multilinear polynomials $mathbb{R}^n o mathbb{R}$ which are annihilated by the differential operator $sum_i partial/partial x_i$. As an application of the last point of view, we show how to lift low-degree functions from a slice to the entire Boolean hypercube while maintaining properties such as expectation, variance and $L^2$-norm.
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机译：我们基于对称群的Young正交表示，为Johnson和Kneser图提供了一个简单，明确的特征向量正交基础。我们的基础也可以看作是布尔超立方体（一片形式为 {（x_1， ldots，x_n） in {0,1 }中的集合）上所有函数的矢量空间的正交基础。 ^ n： sum_i x_i = k } $），以细化Johnson关联方案的本征空间;对于任何可交换的度量，我们的基础都是正交的。更具体地，我们的基础是所有由微分算子$ sum_i partial / partial x_i $消除的所有多项式多项式$ mathbb {R} ^ n to mathbb {R} $的正交基础。作为最后一个观点的应用，我们展示了如何在保持期望，方差和$ L ^ 2 $-范数等属性的同时，将低次函数从切片提升到整个布尔超立方体。
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