---
template: oldpost
title: AMS review of 'Double Macdonald polynomials as the stable limit of Macdonald superpolynomials' by Blondeau-Fournier, Lapointe and Mathieu
date: 2015-07-15
comments: true
archive: false
---
A Macdonald superpolynomial (introduced in \[O. Blondeau-Fournier et
al., Lett. Math. Phys. <span class="bf">101</span> (2012), no. 1, 27–47;
[MR2935476](http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=2935476&loc=fromrevtext);
J. Comb. <span class="bf">3</span> (2012), no. 3, 495–561;
[MR3029444](http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=3029444&loc=fromrevtext)\])
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 \[F. Bergeron and A. M. Garsia, in *Algebraic methods and \\(q\\)-special functions* (Montréal, QC, 1996), 1–52, CRM Proc. Lecture
    Notes, 22, Amer. Math. Soc., Providence, RI, 1999;
    [MR1726826](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=1726826&loc=fromrevtext)\].
    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.