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Grigori Olshanski

Publications and source records attributed to Grigori Olshanski.

At least 19 recordsLinked to original sources
arXiv2026

Hall-Littlewood-positive harmonic functionals on the algebra of symmetric functions

We study the problem of describing the set of real functionals on the quotient $\textrm{Sym}/(p_2-1)$ of the ring of symmetric functions that are nonnegative on the images of certain modified Hall-Littlewood symmetric functions. This question is equivalent to the problem, posed in [Adv Math 395, p.108087 (2022)], of describing the set of coadjoint-invariant measures for unitary groups over a finite field in the infinite-dimensional setting. Our main results constitute partial progress towards this problem. Firstly, we show that the desired set of functionals is very large, in the sense that it contains explicit families of examples depending on infinitely many parameters. Secondly, we provide an analogue of Kerov's mixing construction that produces new sought after functionals from known old ones. This construction depends on an explicit "$p_2$-twisted action" of $\textrm{Sym}$ on itself and the resulting dual map that makes $\textrm{Sym}$ into a comodule. Finally, our third main result explains the relation between the $p_2$-twisted comultiplication and the usual comultiplication on $\textrm{Sym}$.

math.CO↗
arXiv2025

Limits of group algebras for growing symmetric groups and wreath products

Let $S(\infty)$ denote the infinite symmetric group formed by the finitary permutations of the set of natural numbers; this is a countable group. We introduce its virtual group algebra, a completion of the conventional group algebra $\mathbb C[S(\infty)]$. The virtual group algebra is obtained by taking large-$n$ limits of the finite-dimensional group algebras $\mathbb C[S(n)]$ in the so-called tame representations of $S(\infty)$. We establish a connection with the centralizer construction of Molev-Olshanski [J. Algebra, 237 (2001), 302-341; arXiv:math/0002165] and Drinfeld-Lusztig degenerate affine Hecke algebras. This makes it possible to describe the structure of the virtual group algebra. Then we extend the results to wreath products $G\wr S(\infty)$ with arbitrary finite groups $G$.

math.RT↗
arXiv2025

Infinite-dimensional $q$-Jacobi Markov processes

The classical Jacobi polynomials on the interval $[-1,1]$ are eigenfunctions of a second order differential operator. It is well known that this operator generates a diffusion process on $[-1,1]$. Further, this fact admits an extension to $N$ dimensions (Demni (2010), Remling-Rösler (2011)) leading to a $3$-parameter family of diffusion processes $X_N$ on the space of $N$-particle configurations in $[-1,1]$. The generators of the processes $X_N$ are related to Heckman-Opdam's Jacobi polynomials attached to the root system $BC_N$. The first result of the paper shows that the processes $X_N$ have a $q$-analog, the $N$-dimensional $q$-Jacobi processes. These are Feller Markov processes related to the $N$-variate symmetric big $q$-Jacobi polynomials. The later polynomials were introduced and studied by Stokman (1997) and Stokman-Koornwinder (1997); they depend on two Macdonald parameters $(q,t)$ and $4$ extra continuous parameters. The $N$-dimensional $q$-Jacobi processes are still defined on a space of $N$-particle configurations, only now the particles live not on $[-1,1]$ but on certain one-dimensional $q$-grids. The second result (the main one) asserts that the $N$-dimensional $q$-Jacobi processes survive a limit transition as $N$ goes to infinity and two of the extra parameters vary together with $N$ in a certain way. In the limit, one obtains a family of Feller Markov processes which are infinite-dimensional in the sense that they live on configurations with infinitely many particles. The proof uses a lifting of the multivariate big $q$-Jacobi polynomials to the algebra of symmetric functions -- a construction that does not hold for the Heckman-Opdam's Jacobi polynomials. Note also that the large-$N$ limit transition is carried out without any space scaling, which would be impossible in the continuous case.

math.PR↗
arXiv2023

Double Poisson brackets and involutive representation spaces

Let $\Bbbk$ be an algebraically closed field of characteristic $0$ and $A$ be a finitely generated associative $\Bbbk$-algebra, in general noncommutative. One assigns to $A$ a sequence of commutative $\Bbbk$-algebras $\mathcal{O}(A,d)$, $d=1,2,3,\dots$, where $\mathcal{O}(A,d)$ is the coordinate ring of the space $\operatorname{Rep}(A,d)$ of $d$-dimensional representations of the algebra $A$. A double Poisson bracket on $A$ in the sense of Van den Bergh [Trans. Amer. Math. Soc. (2008); arXiv:math/0410528] is a bilinear map $\{\!\!\{-,-\}\!\!\}$ from $A\times A$ to $A^{\otimes 2}$, subject to certain conditions. Van den Bergh showed that any such bracket $\{\!\!\{-,-\}\!\!\}$ induces Poisson structures on all algebras $\mathcal{O}(A,d)$. We propose an analog of Van den Bergh's construction, which produces Poisson structures on the coordinate rings of certain subspaces of the representation spaces $\operatorname{Rep}(A,d)$. We call these subspaces the involutive representation spaces. They arise by imposing an additional symmetry condition on $\operatorname{Rep}(A,d)$ -- just as the classical groups from the series B, C, D are obtained from the general linear groups (series A) as fixed point sets of involutive automorphisms.

math.QA↗
arXiv2023

The centralizer construction and Yangian-type algebras

Let $d$ be a positive integer. The Yangian $Y_d=Y(\mathfrak{gl}(d,\mathbb C))$ of the general linear Lie algebra $\mathfrak{gl}(d,\mathbb C)$ has countably many generators and quadratic-linear defining relations, which can be packed into a single matrix relation using the Yang matrix -- the famous RTT presentation. Alternatively, $Y_d$ can be built from certain centralizer subalgebras of the universal enveloping algebras $U(\mathfrak{gl}(N,\mathbb C))$, with the use of a limit transition as $N\to\infty$. This approach is called the \emph{centralizer construction}. The paper shows that a generalization of the centralizer construction leads to a new family $\{Y_{d,L}: L=1,2,3,\dots\}$ of Yangian-type algebras (the Yangian $Y_d$ being the first term of this family). For the new algebras, the RTT presentation seems to be missing, but a number of properties of the Yangian $Y_d$ persist. In particular, $Y_{d,L}$ possesses a system of quadratic-linear defining relations. The algebras $Y_{d,L}$ ($d=1,2,3,\dots$) provide a kind of quantization for a special double Poisson bracket (in the sense of Van den Bergh) on the free associative algebra with $L$ generators.

math.RT↗
arXiv2023

Remarks on Yangian-type algebras and double Poisson brackets

This short note is an announcement of results. We continue the study of Yangian-type algebras initiated in the paper arXiv:2208.04809. These algebras share a number of properties of the Yangians of type A but are more massive. We refine and substantially enlarge the construction of that paper. A direct link with the class of linear double Poisson brackets on the free associative algebras (Pichereau and Van de Weyer, arXiv:math/0701837) is also established.

math.RT↗
arXiv2023

Characters of classical groups, Schur-type functions, and discrete splines

We study a spectral problem related to the finite-dimensional characters of the groups $Sp(2N)$, $SO(2N+1)$, and $SO(2N)$, which form the classical series $C$, $B$, and $D$, respectively. The irreducible characters of these three series are given by $N$-variate symmetric polynomials. The spectral problem in question consists in the decomposition of the characters after their restriction to the subgroups of the same type but smaller rank $K<N$. The main result of the paper is the derivation of explicit determinantal formulas for the coefficients in this decomposition. In fact, we first compute these coefficients in a greater generality -- for the multivariate symmetric Jacobi polynomials depending on two continuous parameters. Next, we show that the formulas can be drastically simplified for the three special cases of Jacobi polynomials corresponding to the $C$-$B$-$D$ characters. In particular, we show that then the coefficients are given by piecewise polynomial functions. This is where a link with discrete splines arises. In type $A$ (that is, for the characters of the unitary groups $U(N)$), similar results were earlier obtained by Alexei Borodin and the author [Adv. Math., 2012], and then reproved by another method by Leonid Petrov [Moscow Math. J., 2014]. The case of the symplectic and orthogonal characters is more intricate.

math.RT↗
arXiv2022

Mackey-type identity for invariant functions on Lie algebras of finite unitary groups and an application

The Mackey-type identity mentioned in the title relates the operations of parabolic induction and restriction for invariant functions on the Lie algebras of the finite unitary groups $U(N, q^2)$. This result is applied to constructing positive harmonic functions on a new branching graph with a negative Hall-Littlewood parameter, as introduced in the authors' paper [Adv. Math. vol. 395 (2022), 108087; arXiv:2102.01947].This in turn implies the existence of an infinite-parameter family of invariant measures for the coadjoint action of an infinite-dimensional analogue of the groups $U(N, q^2)$.

math.RT↗
arXiv2021

Infinite-dimensional groups over finite fields and Hall-Littlewood symmetric functions

The groups mentioned in the title are certain matrix groups of infinite size over a finite field $\mathbb F_q$. They are built from finite classical groups and at the same time they are similar to reductive $p$-adic Lie groups. In the present paper, we initiate the study of invariant measures for the coadjoint action of these infinite-dimensional groups. We examine first the group $\mathbb{GLB}$, a topological completion of the inductive limit group $\varinjlim GL(n, \mathbb F_q)$. As was shown by Gorin, Kerov, and Vershik [arXiv:1209.4945], the traceable factor representations of $\mathbb{GLB}$ admit a complete classification, achieved in terms of harmonic functions on the Young graph $\mathbb Y$. We show that there exists a parallel theory for ergodic coadjoint-invariant measures, which is linked with a deformed version of harmonic functions on $\mathbb Y$. Here the deformation means that the edges of $\mathbb Y$ are endowed with certain formal multiplicities coming from the simplest version of Pieri rule (multiplication by the first power sum $p_1$) for the Hall-Littlewood (HL) symmetric functions with parameter $t:=q^{-1}$. This fact serves as a prelude to our main results, which concern topological completions of two inductive limit groups built from finite unitary groups. We show that in this case, coadjoint-invariant measures are linked to some new branching graphs. The latter are still related to the HL functions, but the novelty is that now the formal edge multiplicities come from the multiplication by $p_2$ (not $p_1$) and the HL parameter $t$ turns out to be negative (as in Ennola's duality).

math.RT↗
arXiv2021

Macdonald polynomials and extended Gelfand-Tsetlin graph

Using Okounkov's $q$-integral representation of Macdonald polynomials we construct an infinite sequence $Ω_1,Ω_2,Ω_3,\dots$ of countable sets linked by transition probabilities from $Ω_N$ to $Ω_{N-1}$ for each $N=2,3,\dots$. The elements of the sets $Ω_N$ are the vertices of the extended Gelfand-Tsetlin graph, and the transition probabilities depend on the two Macdonald parameters, $q$ and $t$. These data determine a family of Markov chains, and the main result is the description of their entrance boundaries. This work has its origin in asymptotic representation theory. In the subsequent paper, the main result is applied to large-$N$ limit transition in $(q,t)$-deformed $N$-particle beta-ensembles.

math.PR↗
arXiv2020

Determinantal point processes and fermion quasifree states

Determinantal point processes are characterized by a special structural property of the correlation functions: they are given by minors of a correlation kernel. However, unlike the correlation functions themselves, this kernel is not defined intrinsically, and the same determinantal process can be generated by many different kernels. The non-uniqueness of a correlation kernel causes difficulties in studying determinantal processes. We propose a formalism which allows to find a distinguished correlation kernel under certain additional assumptions. The idea is to exploit a connection between determinantal processes and quasifree states on CAR, the algebra of canonical anticommutation relations. We prove that the formalism applies to discrete N-point orthogonal polynomial ensembles and to some of their large-N limits including the discrete sine process and the determinantal processes with the discrete Hermite, Laguerre, and Jacobi kernels investigated by Alexei Borodin and the author in [Commun. Math. Phys. 353 (2017), 853-903; arXiv:1608.01564]. As an application we resolve the equivalence/disjointness dichotomy for some of those processes.

math.PR↗
arXiv2019

The Elliptic Tail Kernel

We introduce and study a new family of $q$-translation-invariant determinantal point processes on the two-sided $q$-lattice. We prove that these processes are limits of the $q$-$zw$ measures, which arise in the $q$-deformation of harmonic analysis on $U(\infty)$, and express their correlation kernels in terms of Jacobi theta functions. As an application, we show that the $q$-$zw$ measures are diffuse. Our results also hint at a link between the two-sided $q$-lattice and rows/columns of Young diagrams.

math-ph↗
arXiv2019

A hierarchy of Palm measures for determinantal point processes with gamma kernels

The gamma kernels are a family of projection kernels $K^{(z,z')}=K^{(z,z')}(x,y)$ on a doubly infinite $1$-dimensional lattice. They are expressed through Euler's gamma function and depend on two continuous parameters $z,z'$. The gamma kernels initially arose from a model of random partitions via a limit transition. On the other hand, these kernels are closely related to unitarizable representations of the Lie algebra $\mathfrak{su}(1,1)$. Every gamma kernel $K^{(z,z')}$ serves as a correlation kernel for a determinantal measure $M^{(z,z')}$, which lives on the space of infinite point configurations on the lattice. We examine chains of kernels of the form $$ \ldots, K^{(z-1,z'-1)}, \; K^{(z,z')},\; K^{(z+1,z'+1)}, \ldots, $$ and establish the following hierarchical relations inside any such chain: Given $(z,z')$, the kernel $K^{(z,z')}$ is a one-dimensional perturbation of (a twisting of) the kernel $K^{(z+1,z'+1)}$, and the one-point Palm distributions for the measure $M^{(z,z')}$ are absolutely continuous with respect to $M^{(z+1,z'+1)}$. We also explicitly compute the corresponding Radon-Nikodým derivatives and show that they are given by certain normalized multiplicative functionals.

math.PR↗
arXiv2018

Interpolation Macdonald polynomials and Cauchy-type identities

Let Sym denote the algebra of symmetric functions and $P_μ(\,\cdot\,;q,t)$ and $Q_μ(\,\cdot\,;q,t)$ be the Macdonald symmetric functions (recall that they differ by scalar factors only). The $(q,t)$-Cauchy identity $$ \sum_μP_μ(x_1,x_2,\dots;q,t)Q_μ(y_1,y_2,\dots;q,t)=\prod_{i,j=1}^\infty\frac{(x_iy_jt;q)_\infty}{(x_iy_j;q)_\infty} $$ expresses the fact that the $P_μ(\,\cdot\,;q,t)$'s form an orthogonal basis in Sym with respect to a special scalar product $\langle\,\cdot\,,\,\cdot\,\rangle_{q,t}$. The present paper deals with the inhomogeneous \emph{interpolation} Macdonald symmetric functions $$ I_μ(x_1,x_2,\dots;q,t)=P_μ(x_1,x_2,\dots;q,t)+\text{lower degree terms}. $$ These functions come from the $N$-variate interpolation Macdonald polynomials, extensively studied in the 90's by Knop, Okounkov, and Sahi. The goal of the paper is to construct symmetric functions $J_μ(\,\cdot\,;q,t)$ with the biorthogonality property $$ \langle I_μ(\,\cdot\,;q,t), J_ν(\,\cdot\,;q,t)\rangle_{q,t}=δ_{μν}. $$ These new functions live in a natural completion of the algebra Sym. As a corollary one obtains a new Cauchy-type identity in which the interpolation Macdonald polynomials are paired with certain multivariate rational symmetric functions. The degeneration of this identity in the Jack limit is also described.

math.CO↗
arXiv2018

The topological support of the z-measures on the Thoma simplex

The Thoma simplex $Ω$ is an infinite-dimensional space, a kind of dual object to the infinite symmetric group. The z-measures are a family of probability measures on $Ω$ depending on three continuous parameters. One of them is the parameter of the Jack symmetric functions, and in the limit when it goes to $0$, the z-measures turn into the Poisson-Dirichlet distributions. The definition of the z-measures is somewhat implicit. We show that the topological support of any nondegenerate z-measure is the whole space $Ω$. The proof is based on results of arXiv:0902.3395 and arXiv:1806.07454.

math.PR↗
arXiv2018

Elements of the q-Askey scheme in the algebra of symmetric functions

The classical q-hypergeometric orthogonal polynomials are assembled into a hierarchy called the q-Askey scheme. At the top of the hierarchy, there are two closely related families, the Askey-Wilson and q-Racah polynomials. As it is well known, their construction admits a generalization leading to remarkable orthogonal symmetric polynomials in several variables. We construct an analogue of the multivariable q-Racah polynomials in the algebra of symmetric functions. Next, we show that our q-Racah symmetric functions can be degenerated into the big q-Jacobi symmetric functions, introduced in a recent paper by the second author. The latter symmetric functions admit further degenerations leading to new symmetric functions, which are analogues of q-Meixner and Al-Salam--Carlitz polynomials. Each of the four families of symmetric functions (q-Racah, big q-Jacobi, q-Meixner, and Al-Salam--Carlitz) forms an orthogonal system of functions with respect to certain measure living on a space of infinite point configurations. The orthogonality measures of the four families are of independent interest. We show that they are linked by limit transitions which are consistent with the degenerations of the corresponding symmetric functions.

math.CO↗
arXiv2017

An analogue of big q-Jacobi polynomials in the algebra of symmetric functions

The main result of the paper is a construction of a five-parameter family of new bases in the algebra of symmetric functions. These bases are inhomogeneous and share many properties of systems of orthogonal polynomials on an interval of the real line. This means, in particular, that the algebra of symmetric functions is embedded into the algebra of continuous functions on a certain compact space Omega, and under this realization, our bases turn into orthogonal bases of weighted Hilbert spaces corresponding to certain probability measures on Omega. These measures are of independent interest --- they are an infinite-dimensional analogue of the multidimensional q-Beta distributions. Our construction uses the big q-Jacobi polynomials and an extension of the Knop-Okounkov-Sahi multivariate interpolation polynomials to the case of infinite number of variables.

math.CO↗