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| author | Yuchen Pei <me@ypei.me> | 2021-06-18 12:58:44 +1000 |
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| committer | Yuchen Pei <me@ypei.me> | 2021-06-18 12:58:44 +1000 |
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| tree | 3127395250cb958f06a98b86f73e77658150b43c /site-from-md/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.html | |
| parent | 4fa26fec8b7e978955e5630d3f820ba9c53be72c (diff) | |
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diff --git a/site-from-md/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.html b/site-from-md/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.html new file mode 100644 index 0000000..71ee1b9 --- /dev/null +++ b/site-from-md/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.html @@ -0,0 +1,61 @@ +<!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> + <!DOCTYPE html> +<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang=""> +<head> + <meta charset="utf-8" /> + <meta name="generator" content="pandoc" /> + <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> + <title>Untitled</title> + <style> + code{white-space: pre-wrap;} + span.smallcaps{font-variant: small-caps;} + span.underline{text-decoration: underline;} + div.column{display: inline-block; vertical-align: top; width: 50%;} + </style> + <!--[if lt IE 9]> + <script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script> + <![endif]--> +</head> +<body> +<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&s1=2935476&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&s1=3029444&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&pg1=MR&s1=1726826&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> +</body> +</html> + + </div> + </div> + </body> +</html> |