AMS review of 'Double Macdonald polynomials as the stable limit of Macdonald superpolynomials' by Blondeau-Fournier, Lapointe and Mathieu

Posted on 2015-07-15

A Macdonald superpolynomial (introduced in [O. Blondeau-Fournier et al., Lett. Math. Phys. 101 (2012), no. 1, 27–47; MR2935476; J. Comb. 3 (2012), no. 3, 495–561; MR3029444]) in \(N\) Grassmannian variables indexed by a superpartition \(\Lambda\) is said to be stable if \({m (m + 1) \over 2} \ge |\Lambda|\) and \(N \ge |\Lambda| - {m (m - 3) \over 2}\) , where \(m\) is the fermionic degree. A stable Macdonald superpolynomial (corresponding to a bisymmetric polynomial) is also called a double Macdonald polynomial (dMp). The main result of this paper is the factorisation of a dMp into plethysms of two classical Macdonald polynomials (Theorem 5). Based on this result, this paper

shows that the dMp has a unique decomposition into bisymmetric monomials;
calculates the norm of the dMp;
calculates the kernel of the Cauchy-Littlewood-type identity of the dMp;
shows the specialisation of the aforementioned factorisation to the Jack, Hall-Littlewood and Schur cases. One of the three Schur specialisations, denoted as \(s_{\lambda, \mu}\), also appears in (7) and (9) below;
defines the \(\omega\) -automorphism in this setting, which was used to prove an identity involving products of four Littlewood-Richardson coefficients;
shows an explicit evaluation of the dMp motivated by the most general evaluation of the usual Macdonald polynomials;
relates dMps with the representation theory of the hyperoctahedral group \(B_n\) via the double Kostka coefficients (which are defined as the entries of the transition matrix from the bisymmetric Schur functions \(s_{\lambda, \mu}\) to the modified dMps);
shows that the double Kostka coefficients have the positivity and the symmetry property, and can be written as sums of products of the usual Kostka coefficients;
defines an operator \(\nabla^B\) as an analogue of the nabla operator \(\nabla\) introduced in [F. Bergeron and A. M. Garsia, in Algebraic methods and \(q\)-special functions (Montréal, QC, 1996), 1–52, CRM Proc. Lecture Notes, 22, Amer. Math. Soc., Providence, RI, 1999; MR1726826]. The action of \(\nabla^B\) on the bisymmetric Schur function \(s_{\lambda, \mu}\) yields the dimension formula \((h + 1)^r\) for the corresponding representation of \(B_n\) , where \(h\) and \(r\) are the Coxeter number and the rank of \(B_n\) , in the same way that the action of \(\nabla\) on the \(n\) th elementary symmetric function leads to the same formula for the group of type \(A_n\) .

Copyright notice: This review is published at http://www.ams.org/mathscinet-getitem?mr=3306078, its copyright owned by the AMS.

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