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diff --git a/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.md b/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.md deleted file mode 100644 index efdd416..0000000 --- a/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.md +++ /dev/null @@ -1,73 +0,0 @@ ---- -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. |