We define lowest weight polynomials (LWPs), motivated by so(d, 2) representation theory, as elements of the polynomial ring over d × n variables obeying a system of first and second order partial differential equations. LWPs invariant under Sn correspond to primary fields in free scalar field theory in d dimensions, constructed from n fields. The LWPs are in one-to-one correspondence with a quotient of the polynomial ring in d × (n − 1) variables by an ideal generated by n quadratic polynomials. The implications of this description for the counting and construction of primary fields are described: an interesting binomial identity underlies one of the construction algorithms. The product on the ring of LWPs can be described as a commutative star product. The quadratic algebra of lowest weight polynomials has a dual quadratic algebra which is non-commutative. We discuss the possible physical implications of this dual algebra.
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de Mello Koch, R., Ramgoolam, S. Free field primaries in general dimensions: counting and construction with rings and modules. J. High Energ. Phys. 2018, 88 (2018). https://doi.org/10.1007/JHEP08(2018)088
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