Coincident root loci and Jack and Macdonald polynomials for special values of the parameters

Abstract

We consider the coincident root loci consisting of the polynomials with at least two double roots and present a linear basis of the corresponding ideal in the algebra of symmetric polynomials in terms of the Jack polynomials with special value of parameter α = −2. As a corollary we present an explicit formula for the Hilbert-Poincarè series of this ideal and the generator of the minimal degree as a special Jack polynomial.A generalization to the case of the symmetric polynomials vanishing on the double shifted diagonals and the Macdonald polynomials specialized at t 2 q = 1 is also presented. We also give similar results for the interpolation Jack polynomials.

Generalized clustering conditions of Jack polynomials at negative Jack parameterα

“…Since the L − operator does not change the value of N ⌽ , maintains the clustering property, and implements the angular momentum lowering, we easily count the polynomials as forming the l = l z max multiplet. For the k =1, r =2, s = 2 case, this counting coincides with the empirical counting observed by Kasatani et al 8 We remark that the expansion of ͑L − ͒ m J k,r,s 0 ␣ k,r in the Jack polynomial basis J ␣ k,r contains ill-behaved Jacks, which diverge at the negative ␣ k,r used. However, their coefficients in the expansion also vanish to give an overall finite contribution.…”

“…However, their coefficients in the expansion also vanish to give an overall finite contribution. These are the "modified" Jacks introduced by Kasatani et al 8 for the specific ͑k , r , s͒ = ͑1,2,2͒ case of the problem studied here. As we reach higher k and s integers, the number of modified Jacks that appear in the expansion of ͑L − ͒ m J k,r,s 0 ␣ k,r grows larger.…”

Generalised discriminants, deformed Calogero–Moser–Sutherland operators and super-Jack polynomials

“…They showed that it is also spanned by the Jack polynomials but the geometry of the corresponding Young diagrams is much more complicated. This important paper shows that the case of special values of y is actually very interesting and deserves more investigation (see [9] for the latest development in this direction). As we have already mentioned to describe the algebraic equations of the discriminant strata is a classical problem which is still largely open [2,4,27].…”

“…well-known book by Gelfand et al [7]) but an alternative term ''generalised coincident root loci'' looks too long and not much better. We would like to mention that the problem of finding the algebraic equations defining the strata in the discriminants is nontrivial and goes back to Arthur Cayley [2] (see [4,9,27] for the recent results in this direction).…”

Deformed Macdonald-Ruijsenaars Operators and Super Macdonald Polynomials

“…The case k = 2, m = 2 was studied in [14]. 15 We investigate the homomorphism: f → M f n,m,q,t in more detail in the next section, but here we finish with the following result mentioned at the beginning of this section.…”