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# Copyright (C) 2013-2021 Yuchen Pei.
# Permission is granted to copy, distribute and/or modify this
# document under the terms of the GNU Free Documentation License,
# Version 1.3 or any later version published by the Free Software
# Foundation; with no Invariant Sections, no Front-Cover Texts, and
# no Back-Cover Texts. You should have received a copy of the GNU
# Free Documentation License. If not, see <https://www.gnu.org/licenses/>.
# This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.
#+title: AMS review of 'Double Macdonald polynomials as the stable limit
#+title: of Macdonald superpolynomials' by Blondeau-Fournier, Lapointe and
#+title: Mathieu
#+date: <2015-07-15>
A Macdonald superpolynomial (introduced in [O. Blondeau-Fournier et al.,
Lett. Math. Phys. 101 (2012), no. 1, 27--47;
[[http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=2935476&loc=fromrevtext][MR2935476]];
J. Comb. 3 (2012), no. 3, 495--561;
[[http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=3029444&loc=fromrevtext][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;
[[http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=1726826&loc=fromrevtext][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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