Related papers: The Selberg integral and Young books
Several methods of evaluation are presented for a family of Selberg-like integrals that arose in the computation of the algebraic-geometric degrees of a family of multiplicity-free nilpotent K_C-orbits. First, adapting the technique of…
A specialisation of a transformation formula for multi-dimensional elliptic hypergeometric series is used to provide compact, non-determinantal formulae for the generating function with respect to the major index of standard Young tableaux…
The number of standard Young tableaux of a skew shape $\lambda/\mu$ can be computed as a sum over excited diagrams inside $\lambda$. Excited diagrams are in bijection with certain lozenge tilings, with flagged semistandard tableaux and also…
The Selberg correlation integrals are averages of the products $\prod_{s=1}^m\prod_{l=1}^n (x_s - z_l)^{\mu_s}$ with respect to the Selberg density. Our interest is in the case $m=1$, $\mu_1 = \mu$, when this corresponds to the $\mu$-th…
The expectation of the descent number of a random Young tableau of a fixed shape is given, and concentration around the mean is shown. This result is generalized to the major index and to other descent functions. The proof combines…
A new q-binomial theorem for Macdonald polynomials is employed to prove an A_n analogue of the celebrated Selberg integral. This confirms the g=A_n case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral…
The Selberg integral, an $n$-dimensional generalization of the Euler beta integral, plays a central role in random matrix theory, Calogero--Sutherland quantum many body systems, Knizhnik--Zamolodchikov equations, and multivariable…
For distinct complex numbers $z_1,...,z_{2N}$, we give a polynomial $P(y_1,...,y_{2N})$ in the variables $y_1,...,y_{2N}$, which is homogeneous of degree $N$, linear with respect to each variable, $sl_2$-invariant with respect to a natural…
Given a direct sum $A$ of full matrix algebras, if there is a combinatorial interpretation associated with both the dimension of $A$ and the dimensions of the irreducible $A$-modules, then this can be thought of as providing an analogue of…
We introduce and study some affine Hecke algebras of type ADE, generalising the affine Hecke algebras of GL. We construct irreducible calibrated representations and describe the calibrated spectrum. This is done in terms of new families of…
In symmetric groups, studies of permutation factorizations or triples of permutations satisfying certain conditions have a long history. One particular interesting case is when two of the involved permutations are long cycles, for which…
We compute the limit shapes of the Young diagrams of the minimal difference $p$ partitions and provide a simple physical interpretation for the limit shapes. We also calculate the asymptotic distribution of the largest part of the Young…
We study a new Selberg-type integral with $n+m$ indeterminates, which turns out to be related to the deformed Calogero-Sutherland systems. We show that the integral satisfies a holonomic system of $n+m$ non-symmetric linear partial…
Using a new presentation for partition algebras (J. Algebraic Combin. 37(3):401-454, 2013), we derive explicit combinatorial formulae for the seminormal representations of the partition algebras. These results generalise to the partition…
We give new product formulas for the number of standard Young tableaux of certain skew shapes and for the principal evaluation of the certain Schubert polynomials. These are proved by utilizing symmetries for evaluations of factorial Schur…
In this expository paper we describe a powerful combinatorial formula and its implications in geometry, topology, and algebra. This formula first appeared in the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey discovered…
The Jacobi-Stirling numbers of the first and second kinds were introduced in 2006 in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial…
For a locally compact abelian group $G$, J. L. Taylor (1971) gave a complete characterization of invertible elements in the measure algebra $M(G)$. Using the Fourier-Stieltjes transform, this characterization can be carried out in the…
The Selberg integral has a twin (`the Dotsenko--Fateev integral') of the following form. We replace real variables $x_k$ in the integrand $\prod |x_k|^{\sigma-1}\,|1-x_k|^{\tau-1} \prod|x_k-x_l|^{2\theta}$ of the Selberg integral by complex…
Algebraic Combinatorics originated in Algebra and Representation Theory, studying their discrete objects and integral quantities via combinatorial methods which have since developed independent and self-contained lives and brought us some…