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diff --git a/site/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.html b/site/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.html new file mode 100644 index 0000000..f64f9b6 --- /dev/null +++ b/site/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.html @@ -0,0 +1,40 @@ +<!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> + </head> + <body> + <header> + <span class="logo"> + <a href="../blog.html">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&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> + + </div> + </div> + </body> +</html> |