# Representation Theory, Probability, and Symmetric Functions

August 22-24, 2019 (Thursday-Saturday)
MIT Room 2-449

Conference Registration

## Organizers:

• Alexei Borodin MIT

## Schedule

### Thursday, Aug 22.

 1:00pm - 2:00pm Diaconis Partial Exhangeability and Reinforced Random Walk Abstract There are many extensions of deFinetti's basic theorem about exchangeable processes. Statisticians have developed them under the name 'partial exchangeablity'. I will review this work (which is closely related to the Gibbs states of statistical physics and the work of the Russian school, particularly Olshansky) and focus on a special case, mixtures of Markov Chains. This has seen surprising application to reenforced random walk and the 'hyperbolic sigma model'. It also gave perhaps my most used work because of it's connections to quantifying distances between DNA strings(the B L A S T algorithm). Naturally, there are open problems and many things to do. 2:15pm - 3:15pm Faraut Horn's Problem and Projection of Orbital Measures for Unitary and Pseudounitary Groups Abstract Let $A$ and $B$ be $n \times n$ Hermitian matrices. Assume that the eigenvalues $\alpha_1,...,\alpha_n$ of $A$ are known, as well as the eigenvalues $\beta_1,...,\beta_n$ of $B$. What can be said about the eigenvalues of the sum $C = A + B$ ? This is Horn's problem. The set of matrices $X$ with spectrum {$\alpha_1,...\alpha_n$} is an orbit $\mathcal{O}_\alpha$ for the natural action of the unitary group $U(n)$ on the space of $n \times n$ Hermitian matrices. Assume that the random matrix $X$ is uniformly distributed on $\mathcal{O}_\alpha$, and, independently, the random matrix $Y$ is uniformly distributed on $\mathcal{O}_\beta$. We will present a formula for the joint distribution of the eigenvalues of the sum $Z = X + Y$. This formula involves projection of orbital measures on the subspace of diagonal matrices. We will also consider Horn's problem for pseudoeigenvalues related to the pseudounitary group $U(p, q)$. 3:15pm - 4:00pm Coffee break 4:00pm - 5:00pm Fulman Random Partitions and Hall-Littlewood Polynomials Abstract Two natural and well-studied measures on integer partitions are the uniform measure and the Plancherel measure. In this talk, we study a third measure, which we believe to be of equal importance, but much less studied. This measure arises in p-adic random matrix theory, in random matrix theory over finite fields, and in the Cohen-Lenstra heuristics of number theory. In this talk, we survey combinatorial properties of this measure.

### Friday, Aug 23.

 11:00am - 12:00pm Sodin Nodal Sets of Random Spherical Harmonics Abstract In the talk I will describe what is known and (mostly) unknown about asymptotic statistical topology of zero sets of random spherical harmonics of large degree on the two-dimensional sphere. I will start with basic open questions and then will discuss a non-trivial lower bound for the variance of the number of connected components of the zero set recently obtained with Fedor Nazarov. Our argument can be viewed as, probably, the first (though, modest) rigorous support of the beautiful Bogomolny-Schmit heuristics, which connects the asymptotic nodal counting with a percolation model on the square lattice. 12:00pm - 2:00pm Lunch break 2:00pm - 3:00pm Noumi Elliptic Hypergeometric Integrals Abstract I will report some recent progresses in the study of elliptic hypergeometric integrals of Selberg type on the basis of collaboration with Masahiko Ito. 3:15pm - 4:15pm Sahi Metaplectic Representations, Weyl Group Actions, and Associated Polynomials Abstract We construct certain representations of affine Hecke algebras, which depend on several auxiliary parameters. We refer to these as “metaplectic” representations, and as a direct consequence we obtain a family of “metaplectic” polynomials, which generalizes the well-known Macdonald polynomials. Our terminology is motivated by the fact that if the parameters are specialized to certain Gauss sums, then our construction recovers the Kazhdan-Patterson action on metaplectic forms for GL(n); more generally it recovers the Chinta-Gunnells action on p-parts of Weyl group multiple Dirichlet series. This is joint work with Jasper Stokman and Vidya Venkateswaran.

### Saturday, Aug 24.

 9:00am - 10:00am Cuenca Point Processes of Representation Theoretic Origin Abstract I will talk about certain point processes, called the "BC type Z-measures", with origins in the representation theory of the infinite-dimensional orthogonal and symplectic groups. The main result I will present is the calculation of their correlation functions in terms of Gauss's hypergeometric function. I will also discuss joint work with Grigori Olshanski on q-analogues of the BC type Z-measures, which have origins in the theory of q-hypergeometric orthogonal polynomials. 10:15am - 11:15am Koornwinder A Nonsymmetric Version of Okounkov's BC-type Interpolation Macdonald Polynomials Abstract In 1998 Okounkov introduced BC-type interpolation Macdonald polynomials. These are symmetric Laurent polynomials which are determined, up to a constant factor, by their vanishing on interpolation points which depend on $q$ and two additional parameters $s$ and $t$. He also showed that Macdonald-Koornwinder polynomials can be explicitly expanded in terms of products of two such interpolation polynomials, one in the variable and one in the dual variable. This so-called binomial formula specializes in the one-variable case to the usual $q$-hypergeometric expression for Askey-Wilson polynomials. Furthermore, Okounkov's polynomials allow extra-vanishing, i.e., they vanish not just on the interpolation points, but also on an additional explicit point set. The talk presents recent work joint with Disveld and Stokman (see arXiv:1808.01221) where we introduce a nonsymmetric version of Okounkov's polynomials. These are Laurent polynomials (no longer symmetric) characterized by their vanishing on interpolation points. The symmetric Okounkov polynomials can be expressed as a sum over the Weyl group for $BC_n$ of the nonsymmetric polynomials. The existence proof of the nonsymmetric polynomials is by a nested induction process. There are experimental indications for extra-vanishing of the nonsymmetric polynomials. 11:15am - 12:00pm Coffee break 12:00pm - 1:00pm Retakh Noncommutative Shifted Symmetric Functions Abstract In 1997 A. Okounkov and G. Olshanski introduced a remarkable deformation of algebra of symmetric functions which they called the algebra of shifted symmetric functions. In this talk, based on a joint work with Robert Laugwitz, we introduce a ring of noncommutative shifted symmetric functions based on an integer-indexed sequence of shift parameters. Using generating series and quasideterminants, this multiparameter approach produces deformations of the ring of noncommutative symmetric functions. Shifted versions of ribbon Schur functions are defined and form a basis for the ring. Further, we produce analogues of Jacobi-Trudi and Nagelsbach-Kostka formulas, a duality anti-algebra isomorphism, shifted quasi-Schur functions, and Giambelli's formula in this setup. In addition, an analogue of power sums is provided, satisfying versions of Wronski and Newton formulas. Finally, a realization of these noncommutative shifted symmetric functions as rational functions in noncommuting variables is given. These realizations have a shifted symmetry under exchange of the variables and are well-behaved under extension of the list of variables.

## Schedule

### Thursday, Aug 22.

• 1:00pm - 2:00pm

Diaconis

Partial Exhangeability and Reinforced Random Walk

Abstract

There are many extensions of deFinetti's basic theorem about exchangeable processes. Statisticians have developed them under the name 'partial exchangeablity'. I will review this work (which is closely related to the Gibbs states of statistical physics and the work of the Russian school, particularly Olshansky) and focus on a special case, mixtures of Markov Chains. This has seen surprising application to reenforced random walk and the 'hyperbolic sigma model'. It also gave perhaps my most used work because of it's connections to quantifying distances between DNA strings(the B L A S T algorithm). Naturally, there are open problems and many things to do.

• 2:15pm - 3:15pm

Faraut

Horn's Problem and Projection of Orbital Measures for Unitary and Pseudounitary Groups

Abstract

Let $A$ and $B$ be $n \times n$ Hermitian matrices. Assume that the eigenvalues $\alpha_1,...,\alpha_n$ of $A$ are known, as well as the eigenvalues $\beta_1,...,\beta_n$ of $B$. What can be said about the eigenvalues of the sum $C = A + B$ ? This is Horn's problem. The set of matrices $X$ with spectrum {$\alpha_1,...\alpha_n$} is an orbit $\mathcal{O}_\alpha$ for the natural action of the unitary group $U(n)$ on the space of $n \times n$ Hermitian matrices. Assume that the random matrix $X$ is uniformly distributed on $\mathcal{O}_\alpha$, and, independently, the random matrix $Y$ is uniformly distributed on $\mathcal{O}_\beta$. We will present a formula for the joint distribution of the eigenvalues of the sum $Z = X + Y$. This formula involves projection of orbital measures on the subspace of diagonal matrices. We will also consider Horn's problem for pseudoeigenvalues related to the pseudounitary group $U(p, q)$.

• 3:15pm - 4:00pm

Coffee break

• 4:00pm - 5:00pm

Fulman

Random Partitions and Hall-Littlewood Polynomials

Abstract

Two natural and well-studied measures on integer partitions are the uniform measure and the Plancherel measure. In this talk, we study a third measure, which we believe to be of equal importance, but much less studied. This measure arises in p-adic random matrix theory, in random matrix theory over finite fields, and in the Cohen-Lenstra heuristics of number theory. In this talk, we survey combinatorial properties of this measure.

### Friday, Aug 23.

• 11:00am - 12:00pm

Sodin

Nodal Sets of Random Spherical Harmonics

Abstract

In the talk I will describe what is known and (mostly) unknown about asymptotic statistical topology of zero sets of random spherical harmonics of large degree on the two-dimensional sphere. I will start with basic open questions and then will discuss a non-trivial lower bound for the variance of the number of connected components of the zero set recently obtained with Fedor Nazarov. Our argument can be viewed as, probably, the first (though, modest) rigorous support of the beautiful Bogomolny-Schmit heuristics, which connects the asymptotic nodal counting with a percolation model on the square lattice.

• 12:00pm - 2:00pm

Lunch break

• 2:00pm - 3:00pm

Noumi

Elliptic Hypergeometric Integrals

Abstract

I will report some recent progresses in the study of elliptic hypergeometric integrals of Selberg type on the basis of collaboration with Masahiko Ito.

• 3:15pm - 4:15pm

Sahi

Metaplectic Representations, Weyl Group Actions, and Associated Polynomials

Abstract

We construct certain representations of affine Hecke algebras, which depend on several auxiliary parameters. We refer to these as “metaplectic” representations, and as a direct consequence we obtain a family of “metaplectic” polynomials, which generalizes the well-known Macdonald polynomials.

Our terminology is motivated by the fact that if the parameters are specialized to certain Gauss sums, then our construction recovers the Kazhdan-Patterson action on metaplectic forms for GL(n); more generally it recovers the Chinta-Gunnells action on p-parts of Weyl group multiple Dirichlet series.

This is joint work with Jasper Stokman and Vidya Venkateswaran.

### Saturday, Aug 24.

• 9:00am - 10:00am

Cuenca

Point Processes of Representation Theoretic Origin

Abstract

I will talk about certain point processes, called the "BC type Z-measures", with origins in the representation theory of the infinite-dimensional orthogonal and symplectic groups. The main result I will present is the calculation of their correlation functions in terms of Gauss's hypergeometric function. I will also discuss joint work with Grigori Olshanski on q-analogues of the BC type Z-measures, which have origins in the theory of q-hypergeometric orthogonal polynomials.

• 10:15am - 11:15am

Koornwinder

A Nonsymmetric Version of Okounkov's BC-type Interpolation Macdonald Polynomials

Abstract

In 1998 Okounkov introduced BC-type interpolation Macdonald polynomials. These are symmetric Laurent polynomials which are determined, up to a constant factor, by their vanishing on interpolation points which depend on $q$ and two additional parameters $s$ and $t$. He also showed that Macdonald-Koornwinder polynomials can be explicitly expanded in terms of products of two such interpolation polynomials, one in the variable and one in the dual variable. This so-called binomial formula specializes in the one-variable case to the usual $q$-hypergeometric expression for Askey-Wilson polynomials. Furthermore, Okounkov's polynomials allow extra-vanishing, i.e., they vanish not just on the interpolation points, but also on an additional explicit point set.

The talk presents recent work joint with Disveld and Stokman (see arXiv:1808.01221) where we introduce a nonsymmetric version of Okounkov's polynomials. These are Laurent polynomials (no longer symmetric) characterized by their vanishing on interpolation points. The symmetric Okounkov polynomials can be expressed as a sum over the Weyl group for $BC_n$ of the nonsymmetric polynomials. The existence proof of the nonsymmetric polynomials is by a nested induction process. There are experimental indications for extra-vanishing of the nonsymmetric polynomials.

• 11:15am - 12:00pm

Coffee break

• 12:00pm - 1:00pm

Retakh

Noncommutative Shifted Symmetric Functions

Abstract

In 1997 A. Okounkov and G. Olshanski introduced a remarkable deformation of algebra of symmetric functions which they called the algebra of shifted symmetric functions. In this talk, based on a joint work with Robert Laugwitz, we introduce a ring of noncommutative shifted symmetric functions based on an integer-indexed sequence of shift parameters.

Using generating series and quasideterminants, this multiparameter approach produces deformations of the ring of noncommutative symmetric functions. Shifted versions of ribbon Schur functions are defined and form a basis for the ring. Further, we produce analogues of Jacobi-Trudi and Nagelsbach-Kostka formulas, a duality anti-algebra isomorphism, shifted quasi-Schur functions, and Giambelli's formula in this setup. In addition, an analogue of power sums is provided, satisfying versions of Wronski and Newton formulas. Finally, a realization of these noncommutative shifted symmetric functions as rational functions in noncommuting variables is given. These realizations have a shifted symmetry under exchange of the variables and are well-behaved under extension of the list of variables.