Pieri Integral Formula and Asymptotics of Jack Unitary Characters

We introduce Jack (unitary) characters and prove two kinds of formulas that are suitable for their asymptotics, as the lengths of the signatures that parametrize them go to infinity. The first kind includes several integral representations for Jack characters of one variable. The second identity we prove is the Pieri integral formula for Jack characters which, in a sense, is dual to the well known Pieri rule for Jack polynomials. The Pieri integral formula can also be seen as a functional equation for irreducible spherical functions of virtual Gelfand pairs. As an application of our formulas, we study the asymptotics of Jack characters as the corresponding signatures grow to infinity in the sense of Vershik-Kerov. We prove the existence of a small $\delta > 0$ such that the Jack characters of $m$ variables have a uniform limit on the $\delta$-neighborhood of the $m$-dimensional torus. Our result specializes to a theorem of Okounkov and Olshanski.Comment: 39 pages. v2: revised after the referee's comments. To appear in Selecta Mathematica, New Serie

Linear versus spin: representation theory of the symmetric groups

We relate the linear asymptotic representation theory of the symmetric groups to its spin counterpart. In particular, we give explicit formulas which express the normalized irreducible spin characters evaluated on a strict partition $\xi$ with analogous normalized linear characters evaluated on the double partition $D(\xi)$. We also relate some natural filtration on the usual (linear) Kerov-Olshanski algebra of polynomial functions on the set of Young diagrams with its spin counterpart. Finally, we give a spin counterpart to Stanley formula for the characters of the symmetric groups.Comment: 41 pages. Version 2: new text about non-oriented (but orientable) map

Jack Derangements

For each integer partition $\lambda \vdash n$ we give a simple combinatorial expression for the sum of the Jack character $\theta^\lambda_\alpha$ over the integer partitions of $n$ with no singleton parts. For $\alpha = 1,2$ this gives closed forms for the eigenvalues of the permutation and perfect matching derangement graphs, resolving an open question in algebraic graph theory. A byproduct of the latter is a simple combinatorial formula for the immanants of the matrix $J-I$ where $J$ is the all-ones matrix, which might be of independent interest. Our proofs center around a Jack analogue of a hook product related to Cayley's $\Omega$--process in classical invariant theory, which we call the principal lower hook product

Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures

This paper establishes universal formulas describing the global asymptotics of two different models of discrete $\beta$-ensembles in high, low and fixed temperature regimes. Our results affirmatively answer a question posed by the second author and \'Sniady. We first consider the Jack measures on Young diagrams of arbitrary size, which depend on the inverse temperature parameter $\beta>0$ and specialize to Schur measures when $\beta=2$. We introduce a class of Jack measures of Plancherel-type and prove a law of large numbers and central limit theorem in the three regimes. In each regime, we provide explicit formulas for polynomial observables of the limit shape and Gaussian fluctuations around the limit shape. These formulas have surprising positivity properties and are expressed in terms of weighted lattice paths. We also establish connections between these measures and the work of Kerov-Okounkov-Olshanski on Jack-positive specializations and show that this is a rich class of measures parametrized by the elements in the Thoma cone. Second, we show that the formulas from limits of Plancherel-type Jack measures are universal: they also describe the limit shape and Gaussian fluctuations for the second model of random Young diagrams of a fixed size defined by Jack characters with the approximate factorization property (AFP) studied by the second author and \'Sniady. Finally, we discuss the limit shape in the high/low-temperature regimes and show that, contrary to the continuous case of $\beta$-ensembles, there is a phase transition phenomenon in passing from the fixed temperature regime to the high/low temperature regimes. We note that the relation we find between the two different models of random Young diagrams appears to be new, even in the special case of $\beta=2$ that relates Schur measures to the character measures with the AFP studied by Biane and \'Sniady.Comment: 64 pages, 7 figure