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diff --git a/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.org b/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.org new file mode 100644 index 0000000..cda6967 --- /dev/null +++ b/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.org @@ -0,0 +1,64 @@ +#+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. |