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As is well known, energy levels appearing in the highly degenerate spectra of the AN−1 type of Haldane-Shastry and Polychronakos spin chains can be classified through the motifs, which are characterized by some sequences of the binary digits like ‘0’ and ‘1’. In a similar way, at present we classify all energy levels appearing in the spectra of the BCN type of Polychronakos spin chains with Hamiltonians containing supersymmetric analogue of polarized spin reversal operators. To this end, we show that the BCN type of multivariate super Rogers-Szegö (SRS) polynomials, which at a certain limit reduce to the partition functions of the later type of Polychronakos spin chains, satisfy some recursion relation involving a q-deformation of the elementary supersymmetric polynomials. Subsequently, we use a Jacobi-Trudi like formula to define the corresponding q-deformed super Schur polynomials and derive a novel expression for the BCN type of multivariate SRS polynomials as suitable linear combinations of the q-deformed super Schur polynomials. Such an expression for SRS polynomials leads to a complete classification of all energy levels appearing in the spectra of the BCN type of Polychronakos spin chains through the ‘branched’ motifs, which are characterized by some sequences of integers of the form (δ1,δ2,...,δN−1|l), where δi∈{0,1} and l∈{0,1,...,N}. Finally, we derive an extended boson-fermion duality relation among the restricted super Schur polynomials and show that the partition functions of the BCN type of Polychronakos spin chains also exhibit similar type of duality relation. |
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