diff options
Diffstat (limited to 'posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.md')
| -rw-r--r-- | posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.md | 73 |
1 files changed, 73 insertions, 0 deletions
diff --git a/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.md b/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.md new file mode 100644 index 0000000..efdd416 --- /dev/null +++ b/posts/2015-07-15-double-macdonald-polynomials-macdonald-superpolynomials.md @@ -0,0 +1,73 @@ +--- +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. |