aboutsummaryrefslogtreecommitdiff
path: root/site/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.html
blob: a0d5a7cf6e6b39cec82189376f8014061cb6477e (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
<!doctype html>
<html lang="en">
    <head>
        <meta charset="utf-8">
        <title>AMS review of 'Double Macdonald polynomials as the stable limit of Macdonald superpolynomials' by Blondeau-Fournier, Lapointe and Mathieu</title>
        <link rel="stylesheet" href="../assets/css/default.css" />
        <script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script>
        <script src="../assets/js/analytics.js" type="text/javascript"></script>
    </head>
    <body>
        <header>
            <span class="logo">
                <a href="../blog.html">Yuchen's Blog</a>
            </span>
            <nav>
                <a href="../index.html">About</a><a href="../postlist.html">All posts</a><a href="../blog-feed.xml">Feed</a>
            </nav>
        </header>

        <div class="main">
            <div class="bodyitem">
                <h2> AMS review of 'Double Macdonald polynomials as the stable limit of Macdonald superpolynomials' by Blondeau-Fournier, Lapointe and Mathieu </h2>
                <p>Posted on 2015-07-15</p>
                    <p>A Macdonald superpolynomial (introduced in [O. Blondeau-Fournier et al., Lett. Math. Phys. <span class="bf">101</span> (2012), no. 1, 27–47; <a href="http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&amp;s1=2935476&amp;loc=fromrevtext">MR2935476</a>; J. Comb. <span class="bf">3</span> (2012), no. 3, 495–561; <a href="http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&amp;s1=3029444&amp;loc=fromrevtext">MR3029444</a>]) 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</p>
<ol type="1">
<li><p>shows that the dMp has a unique decomposition into bisymmetric monomials;</p></li>
<li><p>calculates the norm of the dMp;</p></li>
<li><p>calculates the kernel of the Cauchy-Littlewood-type identity of the dMp;</p></li>
<li><p>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;</p></li>
<li><p>defines the \(\omega\) -automorphism in this setting, which was used to prove an identity involving products of four Littlewood-Richardson coefficients;</p></li>
<li><p>shows an explicit evaluation of the dMp motivated by the most general evaluation of the usual Macdonald polynomials;</p></li>
<li><p>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);</p></li>
<li><p>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;</p></li>
<li><p>defines an operator \(\nabla^B\) as an analogue of the nabla operator \(\nabla\) introduced in [F. Bergeron and A. M. Garsia, in <em>Algebraic methods and \(q\)-special functions</em> (Montréal, QC, 1996), 1–52, CRM Proc. Lecture Notes, 22, Amer. Math. Soc., Providence, RI, 1999; <a href="http://www.ams.org/mathscinet/search/publdoc.html?r=1&amp;pg1=MR&amp;s1=1726826&amp;loc=fromrevtext">MR1726826</a>]. 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 \(A_n\) .</p></li>
</ol>
<p>Copyright notice: This review is published at http://www.ams.org/mathscinet-getitem?mr=3306078, its copyright owned by the AMS.</p>

            </div>
        </div>
    </body>
</html>