arXiv subjects

Maciej Dołęga

Publications and source records attributed to Maciej Dołęga.

24 records · Page 2Linked to original sources

arXiv2017

Strong factorization property of Macdonald polynomials and higher-order Macdonald's positivity conjecture

We prove a strong factorization property of interpolation Macdonald polynomials when $q$ tends to $1$. As a consequence, we show that Macdonald polynomials have a strong factorization property when $q$ tends to $1$, which was posed as an open question in our previous paper with Féray. Furthermore, we introduce multivariate $q,t$-Kostka numbers and we show that they are polynomials in $q,t$ with integer coefficients by using the strong factorization property of Macdonald polynomials. We conjecture that multivariate $q,t$-Kostka numbers are in fact polynomials in $q,t$ with nonnegative integer coefficients, which generalizes the celebrated Macdonald's positivity conjecture.

math.CO↗

arXiv2017

Cumulants of Jack symmetric functions and $b$-conjecture

Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $ψ(x, y, z; t, 1+β)$ that might be interpreted as a continuous deformation of the generating series of rooted hypermaps. They made the following conjecture: the coefficients of $ψ(x, y, z; t, 1+β)$ in the power-sum basis are polynomials in $β$ with nonnegative integer coefficients (by construction, these coefficients are rational functions in $β$). We prove partially this conjecture, nowadays called $b$-conjecture, by showing that coefficients of $ψ(x, y, z; t, 1+ β)$ are polynomials in $β$ with rational coefficients. A key step of the proof is a strong factorization property of Jack polynomials when the Jack-deformation parameter $α$ tends to $0$, that may be of independent interest.

math.CO↗

arXiv2016

A bijection for rooted maps on general surfaces

We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, that is orientable and non-orientable as well. This general construction requires new ideas and is more delicate than the special orientable case, but it carries the same information. In particular, it leads to a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable rooted connected maps of Euler characteristic $2-2h$, and of the algebraicity of their generating functions, similar to the one previously obtained in the orientable case via the Marcus-Schaeffer bijection. It also shows that the renormalization factor $n^{1/4}$ for distances between vertices is universal for maps on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation on any fixed surface converge in distribution when the size $n$ tends to infinity. Finally, we extend the Miermont and Ambjørn-Budd bijections to the general setting of all surfaces. Our construction opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.

math.CO↗

arXiv2015

Gaussian fluctuations of Young diagrams and structure constants of Jack characters

In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first and second-order asymptotics of the bulk of a random Young diagram under this distribution, which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the first order asymptotics) and Kerov (for the second order asymptotics). This gives more evidence of the connection with Gaussian $β$-ensemble, already suggested by some work of Matsumoto. Our main tool is a polynomiality result for the structure constant of some quantities that we call Jack characters, recently introduced by Lassalle. We believe that this result is also interested in itself and we give several other applications of it.

math.PR↗

arXiv2014

Jack polynomials and orientability generating series of maps

We study Jack characters, which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization. These quantities have been introduced by Lassalle who formulated some challenging conjectures about them. We conjecture existence of a weight on non-oriented maps (i.e., graphs drawn on non-oriented surfaces) which allows to express any given Jack character as a weighted sum of some simple functions indexed by maps. We provide a candidate for this weight which gives a positive answer to our conjecture in some, but unfortunately not all, cases. In particular, it gives a positive answer for Jack characters specialized on Young diagrams of rectangular shape. This candidate weight attempts to measure, in a sense, the non-orientability of a given map.

math.CO↗

arXiv2014

On Kerov polynomials for Jack characters

We consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We give a partial result in this direction, showing that some quantities are polynomials in the Jack parameter $α$ with prescribed degree. Our result has several interesting consequences in various directions. Firstly, we give a new proof of the fact that the coefficients of Jack polynomials expanded in the monomial or power-sum basis depend polynomially in $α$. Secondly, we describe asymptotically the shape of random Young diagrams under some deformation of Plancherel measure.

math.CO↗