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  "body": "\n## Summary\n\nJack functions $J_\\lambda^{(\\alpha)}$ are [[symmetric functions|PAJEHWXW6M]] indexed by an [[integer partition|0JV30XN73Z]] $\\lambda$ and depending on a parameter $\\alpha$.\nThey are fixed by being triangular with respect to the monomial basis and orthogonal for a deformation of the Hall inner product.\nFor $\\alpha = 1$ they are multiples of the [[Schur polynomials|43VFMJCDYJ]], and for $\\alpha = 2$ they are the zonal polynomials of multivariate statistics.\nThey are the limit $q = t^\\alpha$, $t \\to 1$ of the [[Macdonald polynomials|VYP9PB000X]], and they are eigenfunctions of the Calogero–Sutherland model of quantum particles on a circle.\n\n## Explanations\n\n### L1 human\n\nSchur polynomials are a very good basis of the symmetric polynomials, because they are orthogonal and have a meaning in representation theory.\nJack functions keep the orthogonality but add a dial $\\alpha$ that changes the inner product.\nAs $\\alpha$ moves, the basis moves from the elementary symmetric functions ($\\alpha = 0$) through the Schur functions ($\\alpha = 1$) to the monomial symmetric functions ($\\alpha \\to \\infty$).\nFor example, in degree $2$ the monic Jack function is $m_{(2)} + \\frac{2}{1 + \\alpha} m_{(1,1)}$, which equals $s_{(2)} = m_{(2)} + m_{(1,1)}$ at $\\alpha = 1$.\n",
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