Related papers: AFLT-type Selberg integrals
We show that for Jack parameter \alpha = -(k+1)/(r-1), certain Jack polynomials studied by Feigin-Jimbo-Miwa-Mukhin vanish to order r when k+1 of the coordinates coincide. This result was conjectured by Bernevig and Haldane, who proposed…
By generalizing the classical Selberg-Chowla formula, we establish the analytic continuation and functional equation for a large class of Epstein zeta functions. This continuation is studied in order to provide new classes of theorems…
We construct and study an explicit simultaneous $\mathscr{Y}$ eigenbasis of Ion and Wu's standard representation of the $^+$stable-limit double affine Hecke algebra for the limit Cherednik operators $\mathscr{Y}_i$. This basis arises as a…
In this paper, we introduce a way to generalize the Euler's gamma function as well as some related special functions. With a given polynomial in one variable $f(t)\ge 0$, we can associate a function, so-called "gamma function associated…
Egge, Loehr and Warrington gave in \cite{ELW} a combinatorial formula that permits to convert the expansion of a symmetric function, homogeneous of degree $n$, in terms of Gessel's fundamental quasisymmetric functions into an expansion in…
We introduce a new combinatorial object, semistandard increasing decomposition tableau and study its relation to a semistandard decomposition tableau introduced by Kra\'skiewicz and developed by Lam and Serrano. We also introduce…
We derive precise formulas for the archimedean Euler factors occurring in certain standard Langlands $L$-functions for unitary groups. In the 1980s, Paul Garrett, as well as Ilya Piatetski-Shapiro and Stephen Rallis (independently of…
The solution of Shareshian-Wachs conjecture by Brosnan-Chow linked together the cohomology of regular semisimple Hessenberg varieties and graded chromatic symmetric functions on unit interval graphs. On the other hand, it is known that…
For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of…
In this paper, we study shifted Schur functions $S_\mu^\star$, as well as a new family of shifted symmetric functions $\mathfrak{K}_\mu$ linked to Kostka numbers. We prove that both are polynomials in multi-rectangular coordinates, with…
We continue a program generalizing classical results from the analysis on symmetric cones to the Dunkl setting for root systems of type A. In particular, we prove a Dunkl-Laplace transform identity for Heckman-Opdam hypergeometric functions…
In this paper we calculate some Generalized Selberg integrals. The answer is expressed in terms of $\Gamma$-functions. Integrals of this type serve as normalization constants or directly via undoing 2-D integrals for determination of…
We provide a general construction of integral TQFTs over a general commutative ring, $\mathbf{k}$, starting from a finite Hopf algebra over $\mathbf{k}$ which is Frobenius and double balanced. These TQFTs specialize to the Hennings…
We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum…
By a similar idea for the construction of Milnor's gamma functions, we introduce "higher depth determinants" of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant…
It is well known that if one integrates a Schur function indexed by a partition $\lambda$ over the symplectic (resp. orthogonal) group, the integral vanishes unless all parts of $\lambda$ have even multiplicity (resp. all parts of $\lambda$…
The connection problem associated with a Selberg type integral is solved. The connection coefficients are given in terms of the $q$-Racah polynomials. As an application of the explicit expression of the connection coefficients, examples of…
The aim of this paper is to present a new simple recurrence for Appell and Sheffer sequences in terms of the linear functional that defines them, and to explain how this is equivalent to several well-known characterizations appearing in the…
The Jack polynomials with prescribed symmetry are obtained from the nonsymmetric polynomials via the operations of symmetrization, antisymmetrization and normalization. After dividing out the corresponding antisymmetric polynomial of…
A systematic study of the representation theory of double affine Hecke algebras and related harmonic analysis is started in this paper. Continuing the previous papers we use the technique of intertwining operators to create Macdonald…