Let  be the subring of the polynomial ring of n independent variables generated by the differences Let be the invariant subring of  under the action of the symmetric group  of degree n defined by  We investigate and construct a -basis of the submodules  of  of degree d under certain conditions. Furthermore,  is expressed as the kernel of the differential operator  of the -invariant subring  for elementary symmetric polynomials of degree t of  With respect to an ordered -basis of  we determine the exponents n on the minimal terms of the -basis vectors of  (Theorem A).

invariant subring, symmetric functions, lexicographic order, step matrix

Received: December 10, 2024;  Accepted: March 7, 2025; Published: March 31, 2025

How to cite this article: Takashi Maeda, A -basis of the -invariant subring of  JP Journal of Algebra, Number Theory and Applications 64(3) (2025), 251-287. https://doi.org/10.17654/0972555525015

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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