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Charles F. Dunkl, Singular polynomials and modules for the symmetric groups, International Mathematics Research Notices, Volume 2005, Issue 39, 2005, Pages 2409–2436, https://doi.org/10.1155/IMRN.2005.2409
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Abstract
For certain negative rational numbers κ0, called singular values, and associated with the symmetric group SN on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter κ=κ0. It was shown by de Jeu, Opdam, and the author (1994) that the singular values are the values −m/n with 2 ≤ n ≤ N, m=1,2,\…, and m/n is not an integer. For each pair (m,n) satisfying these conditions, there is a unique irreducible SN-module of singular polynomials. The existence of these polynomials was previously established by the author (2004). The uniqueness is proven in the present paper. By using Murphy's results (1981), the possible existence of singular polynomials is restricted to the isotype τ where τ is a partition of N satisfying the condition that n/gcd(m,n) divides τi+1 for 1 ≤ i < l; l is the length of τ, that is, τl>τl+1=0. By means of nonsymmetric Jack polynomials, it is shown that the assumption τ2≥ n/gcd(m,n) leads to a contradiction. This shows that the singular polynomials are exactly those already determined, and are of isotype τ, where τ2=…=τl−1=(n/gcd(m,n)) −1≥τl.