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

(1) shows that the dMp has a unique decomposition into bisymmetric monomials;

(2) calculates the norm of the dMp;

(3) calculates the kernel of the Cauchy-Littlewood-type identity of the dMp;

(4) 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;

(5) defines the $$\omega$$ -automorphism in this setting, which was used to prove an identity involving products of four Littlewood-Richardson coefficients;

(6) shows an explicit evaluation of the dMp motivated by the most general evaluation of the usual Macdonald polynomials;

(7) 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);

(8) 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;

(9) 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 $$A_n$$ .

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