Imagine two cars, slow (S) and fast (F), driving to the right on a discrete 1-dimensional lattice according to some random walk mechanism, and such that the cars cannot pass each other. We consider two systems, SF and FS, depending on which car is ahead. It is known for some time (through connections to symmetric functions and the RSK correspondence) that if at time 0 the cars are immediate neighbors, the trajectory of the car that is behind is the same (in distribution) in both systems. However, this fact fails when the initial locations of the cars are not immediate neighbors. I will explain how to recover the identity in distribution by suitably randomizing the initial condition in one of the systems. This result arises in our recent work on multiparameter stochastic systems (where the parameters are speeds attached to each car) in which the presence of parameters preserves the quantum integrability. This includes TASEP (totally asymmetric simple exclusion process), its deformations, and stochastic vertex models, which are all integrable through the Yang-Baxter equation (YBE). In the context of car dynamics, we interpret YBEs as Markov operators intertwining the transition semigroups of the dynamics of the processes differing by a parameter swap. We also construct Markov processes on trajectories which "rewrite the history" of the car dynamics, that is, produce an explicit monotone coupling between the trajectories of the systems differing by a parameter swap. Based on a joint work with Axel Saenz.

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Colored bosonic lattice models were studied and applied by Borodin and Wheeler. We will consider particular bosonic models whose theory is strikingly parallel to the fermionic models described by Brubaker, Buciumas, Bump and Gustafsson, who applied them to study Whittakerfunctions of GL_r(F) over a nonarchimedean local field. The many parallels with the BBBG theory include a monochrome factorization, a local lifting property, and very similar R-matrices; indeed the R-matrices are the same except for a very minor change which reflects the quantum group U_q(sl_{r+1}) in the bosonic models, versus U_q(gl(r|1)) in the fermionic models. We will show that the partition functions of the bosonic models are equal to matrix coefficients of principal series representations of GL_r(F), where F is a nonarchimedean local field. We will also explain the relationship with nonsymmetric Hall-Littlewood polynomials. This is joint work with Slava Naprienko.

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We use the free-fermionic six-vertex model to define the double factorial (supersymmetric) Schur functions which generalize and unify classical Schur functions, factorial Schur functions, supersymmetric Schur functions, factorial supersymmetric Schur functions, and dual Schur functions from literature. We show how to use the techniques of the lattice model to give intuitive proofs for Cauchy identity, Jacobi-Trudi Identity, Factorization, Determinant expression, and other familiar results.

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In an effort to provide a bijective version of the proof in Weyl Group Multiple Dirichlet Series - Type A Combinatorial Theory by Brubaker, Bump, and Friedberg, our Polymath Jr. project group introduces a lattice model refinement and conjectures a statistic-preserving bijection between refined lattice model states. We motivate the problem and build up the necessary background to state the conjecture.

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We will review the "usual paradigm" for a class of solvable lattice models including the six-vertex model and its generalizations. This attaches an element of a braided tensor category, such as the module category for a quantum group, to each edge of a grid. Examples found by Buciumas, Brubaker, Bump and Gustafsson related to quantum affine gl(r|n) seem to be slightly outside this paradigm. We will review the Kac modules for gl(r|n) and speculate how they might help with this puzzle. This line of thought leads to the following prediction: the q-Fock space of Kashiwara, Miwa and Stern, defined as a module for quantum affine gl(n), is secretly also a module for quantum affine gl(1|n).

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This expository talk will describe the fusion procedure, a mechanism (introduced by Kulish, Reshetikhin, and Sklyanin in 1981) for producing new solutions to the Yang-Baxter equation starting from an initial one. We will explain how to implement this procedure in the example of fusing fundamental weights of U_q (\hat{sl}_2), and we will outline some of the uses of fusion in algebraic combinatorics and probability.

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Using the asymmetric six-vertex model on the infinite lattice we discuss a generalisation of the boson-fermion correspondence for Macdonald’s characteristic map in the case of the Hecke algebra. The partition function of the six-vertex model yields the irreducible Hecke characters as first described by Ram. Imposing periodic boundary conditions on the lattice we obtain (virtual) Cylindric Hecke characters and show that they span a positive subcoalgebra in the Grothendieck ring of Hecke algebras (seen as Hopf algebra) whose structure constants are the Gromov-Witten invariants of Grassmannians.

This talk is based on C.K. Comm. Math. Phys. 381 (2021) 591

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We introduce a family of Markov growth processes on discrete height functions defined on the 2-dimensional square lattice. Each height function corresponds to a configuration of the six vertex model on the infinite square lattice. We focus on the stochastic six vertex model corresponding to a particular two-parameter family of weights within the ferroelectric regime.

The Markov processes we construct preserve the KPZ pure states in the full plane. We also show that the same processes put on the torus preserve arbitrary Gibbs measures for generic six vertex weights (not necessarily in the ferroelectric regime).

Our dynamics arise naturally from the Yang–Baxter equation for the six vertex model via its bijectivisation. The dynamics we construct are irreversible; in particular, the height function has a nonzero average drift. In each KPZ pure state, we explicitly compute the average drift (also known as the current) as a function of the slope. We use this to analyze the hydrodynamics of a non-stationary version of our process acting on quarter plane stochastic six vertex configurations.

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We study $k$-tilings ($k$-tuples of domino tilings) of the Aztec diamond of rank $m$. We assign a weight to each $k$-tiling, depending on the number of vertical dominos and also on the number of ``interactions" between the different tilings. We will compute the generating polynomials of the $k$-tilings by relating them to an integrable colored vertex model. We will then prove some combinatorial results about $k$-tilings in certain limits of the interaction strength.

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I will review two well-known integrability classes of the six-vertex model and introduce their generalizations.

The field-free class with a1=a2, b1=b2, c1=c2. The partition function is given by the Izergin-Korepin determinant. The homogeneous limit of the determinant gives the enumeration of the alternating sign matrices and refinements. I extend the class to b1 != b2 and show that the generalized Izergin-Korepin determinant holds.

The free-fermionic class with parameter a1a2 + b1b2 = c1c2. The partition function gives the special functions: Schur polynomials, Spherical Whittaker functions, and refinements depending on the choice of weights. I compute the partition function for the generic free-fermionic weights. It has the form of the deformed Weyl denominator and Schur-like function.

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LLT polynomials, originally known as ribbon functions, are a q-analogue of products of Schur polynomials. Sharing many of the same properties as Schur polynomials--they are symmetric, satisfy Cauchy identities, and can be written as a generating function over tableaux as well as in terms of operators on a Fock space representation--they were one of the inspirations for Lam's combinatorial generalization of the Boson-Fermion correspondence (arXiv:0507341).

I will discuss the supersymmetric analogue of LLT polynomials, known as superLLT polynomials, and present a solvable ribbon lattice model that produces them as its partition function. I will also prove the Cauchy identity for these polynomials using operators and compare it to a surprising solution to the Yang-Baxter equation that arises when we adapt the lattice model to stack atop itself in a Cauchy-identity-style model. This is joint work with Michael Curran, Calvin Yost-Wolff, Sylvester Zhang, and Valerie Zhang (arXiv:2110.07597).

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We develop a new point of view on shuffle algebras based on exactly solvable lattice models. We establish in this way a nontrivial isomorphism between the center of the Hecke algebra and the shuffle algebra related to toroidal gl(1). If time permits, we shall give an application to the Hilbert series of the commuting scheme.

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I will discuss random perfect matchings on a large class of graphs called rail yard graphs, such that the perfect matchings form Schur processes. I will then talk about the limit shape (Law of Large Numbers) and convergence of height fluctuations to Gaussian Free Field (Central Limit Theorem) for random perfect matchings on these graphs in the scaling limit. The techniques used include analyzing a large class of Macdonald processes which contain dual partitions as well.

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Dual Grothendieck polynomials are dual to symmetric Grothendieck polynomials under the Hall inner product. In this talk, I will talk about applications of solvable lattice models to dual Grothendieck polynomials and their structure constants. I will also introduce a set of polynomials that are dual to factorial Grothendieck polynomials. This talk is based on joint work with Paul Zinn-Justin and Travis Scrimshaw.

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Note the unusual time!

Limit shape formation is a common feature of highly correlated statistical mechanical systems. On the macroscopic scale the random system settles into a deterministic limit often exhibiting fascinating arctic boundaries. These geometric limit shapes are minimisers of gradient variational problems with a surface tension which encodes the local entropy of the model.

I’ll describe an integrability principle for variational problems in terms of an intrinsic complex variable. Across a variety of models all limit shapes can be parametrised by harmonic functions in this variable. I argue that the integrability of the limit shape PDE is a reflection of the integrability of the model. Some novel features characteristic to non-free fermionic settings will be discussed.

The talk is based on joint works with Rick Kenyon and will complement his talk from last week.

References: Gradient variational problems in R^2, The genus-zero five-vertex model

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This talk is based on joint work with Jan de Gier, Sam Watson, and Istvan Prause.

The five vertex model is a special case of the six-vertex model.

We give a complete solution to the model, and explicit parameterizations of limit shapes such as the ``box-plane partition” limit shape generalizing that for the lozenge tiling case.

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Surface diffusion is the physical phenomenon where particles move across a surface without leaving or arriving on the surface. Consequently, the total number of particles is conserved. We are interested in identifying Markov jump processes that are adequate models for surface diffusion and amenable to computations. In 2014, Y. Takeyama proposed a deformation of affine Hecke algebra that can be represented on Laurent polynomial ring using the multiplication and difference operators of Lascoux and Schutzenberger and on the vector space of complex-valued functions defined on the lattice. Takeyama constructed a Hamiltonian and a model for diffusion across a surface with a one-dimensional substrate using the latter representation. In addition, Takeyama showed that some eigenfunctions of the Hamiltonian could be constructed using Bethe ansatz. We aim to extend Takeyama's model to incorporate movement of the particles in both right and left directions. Results are presented for the case of two particles.

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Macdonald often included in his talks the “specialization square” for symmetric Macdonald polynomials, which has monomial symmetric functions on the top edge, elementary symmetric functions on the right edge, Hall-Littlewood polynomials on the left edge, Schur functions on the diagonal and Jack polynomials in the upper right corner. This talk will explore the analogous “specialization square” for non-symmetric Macdonald polynomials. The primary tools are the monomial expansion formulas, the E-expansion formula, and the creation formula.

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We review several uses of color in the literature of solvable lattice models and track their connections to quantum group modules and to various applications to representation theory and symmetric function theory. My introduction to colored models came through Borodin and Wheeler's work (now a "classical" result, owing to the pace of the field: see arXiv:1808.01866). Inspired by these, with my collaborators and students, we sought other uses of color in the representation theory of metaplectic groups, in Schubert calculus, etc. We will sort out the seemingly confusing sources and applications of color from the above results, paying special attention to the paper arXiv:2007.04310 (with Frechette, Hardt, Tibor, and Weber) on Grothendieck polynomials whose "version 3" was recently revised and generalized in September 2021.

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It is known that Schur symmetric polynomials admit a number of generalizations (Macdonald's 1992 variations) which retain determinantal structure - for example, factorial and supersymmetric Schur functions. We describe an overarching family of Schur-like rational functions arising as partition functions of fully inhomogeneous free fermion six vertex model. These functions are indexed by partitions, have as variables the pairs (x_i,r_i), i=1,...,N, of horizontal rapidities and spin parameters; and, moreover, depend on vertical rapidities and spin parameters (y_j,s_j), j>=1. We establish determinantal formulas, orthogonality, Cauchy identities, and other properties of our functions. We also introduce random domino tiling models based on the Schur rational functions (a la Schur processes of Okounkov-Reshetikhin 2001), and obtain bulk (lattice) asymptotics leading to a new deformation of the extended discrete sine kernel. Based on the joint project https://arxiv.org/abs/2109.06718 with A. Aggarwal, A. Borodin, and M. Wheeler.

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Infinite dimensional analogues of classical formulas from the theory of p-adic groups give rise to a certain correction factor. For example, Macdonald's formula for the spherical function and the Casselman-Shalika formula, when extended to the affine, and general Kac-Moody setting, all have this feature.

We will discuss this correction factor. In affine type, it is known by Cherednik's work on Macdonald's constant term conjecture. More generally, it can be represented as a collection of polynomials of t indexed by positive imaginary roots; these are deformations of root multiplicities. Methods of computing imaginary root multiplicities, such as the Peterson algorithm and the Berman-Moody formula can be generalized to compute the correction factor for any t. They both reveal some properties of the correction factor and raise further questions and conjectures about its structure. This is joint work with Dinakar Muthiah and Ian Whitehead.

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The commutative (trigonometric) shuffle algebra is known to be isomorphic to the ring of symmetric functions (arxiv.org/abs/0904.2291v1). In this isomorphism one writes an integral operator where the shuffle algebra element stands as a kernel and the symmetric function gives the eigenvalues of this operator. The eigenvectors of these integral operators are the Macdonald functions. I will explain this construction and then discuss some applications.

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The Cauchy Identities are fundamental features of a number of families of symmetric functions. For the Schur polynomials Cauchy Identities can be proven bijectively using the RSK correspondence. For Macdonald polynomials producing an elementary proof of the same identities has remained an outstanding challenge.

In this talk I will show how we solve this problem in the case of the q-Whittaker polynomials, i.e. Macdonald polynomials with t parameter set to zero. It turns out that the RSK correspondence can be q-deformed in a bijective fashion by properly lifting its set of symmetries. For this we employ results coming from the theory of Kashiwara's crystals and the bijection will be a result of a novel affine bi-crystal structure respectively on the set of infinite matrices and semi-standard tableaux. Our arguments pivot around a combination of various theories that include Demazure crystals, the Box-Ball system or Sagan and Stanley's skew RSK correspondence.

This is a joint work with Takashi Imamura and Tomohiro Sasamoto and it is a continuation of last week's seminar by Tomohiro Sasamoto.

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References:
[1] Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto, Identity between restricted Cauchy sums for the q-Whittaker and skew Schur polynomals, arXiv: 2106.11913
[2] Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto, Skew RSK dynamics: Greene invariants, affine crystals and applications to q-Whittaker polynomials, arXiv: 2106.11922

The q-Whittaker measure has been playing an important role in integrable probability. It describes the position of a particle in discrete models in the KPZ class such as the q-(Push)TASEP. Though the q-Whittaker measure is not directly associated with a determinantal point process(DPP), the q-Laplace transform of its marginal is written as a Fredholm determinant through a few methods, which allows establishing Tracy-Widom asymptotics.

On the other hand, the periodic Schur measure was first introduced by Borodin in 2007. Its shift mix version is a DPP and its correlation functions can be studied in a standard manner. It is also intimately related to a free fermion at finite temperature.

In our recent works [1,2], we have proved an identity between marginals of the two measures. In [1] the identity was shown by a matching of Fredholm determinants while in [2] it was proved bijectively by generalizing and developing substantially the RSK algorithm, and studying its properties leveraging the theory of affine crystal.

Our identity presents a direct connection between discretized models of the KPZ equation and free fermions at finite temperature, providing a new approach to study KPZ models.

In the first part of the talk, we will give an overview of our new results. Details of the most novel part of our results, namely the bijective proof of the identity, will be explained in the second talk.

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References:
[1] Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto, Identity between restricted Cauchy sums for the q-Whittaker and skew Schur polynomals, arXiv: 2106.11913
[2] Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto, Skew RSK dynamics: Greene invariants, affine crystals and applications to q-Whittaker polynomials, arXiv: 2106.11922

The Airy_2 process is a universal distribution which describes fluctuations in models in the Kardar--Parisi--Zhang (KPZ) universality class, such as the asymmetric simple exclusion process (ASEP) and the Gaussian Unitary Ensemble (GUE). Despite its ubiquity, there are no proven results for analogous fluctuations of multi--species models. Here, we will discuss one model in the KPZ universality class, the $q$--Boson. We will show that the joint multi--point fluctuations of the single--species $q$--Boson match the single--point fluctuations of the multi--species $q$--Boson. Therefore the single--point fluctuations of multi--species models in the KPZ class ought to be the Airy_2 process. The proof utilizes the underlying algebraic structure of the multi--species $q$--Boson, namely the quantum group symmetry and Coxeter group actions.

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Relevant papers: Stochastic duality of ASEP with two particle types via symmetry of quantum groups of rank two, A Multi-species ASEP(q,j) and q-TAZRP with Stochastic Duality, https://arxiv.org/abs/1701.04468, Probability distributions of multi-species q-TAZRP and ASEP as double cosets of parabolic subgroups

I will discuss exact, multiple integral formulas for the transition probability (Green's function) of two different integrable two-species stochastic particle models: the Arndt-Heinzel-Rittenberg (AHR) model and the 2-TASEP whose generator is the $q\rightarrow 0$ limit of the R-matrix related to $U_q(sl(3))$. We derive closed form formulas for total crossing probabilities. In the case of the AHR I will sketch how an asymptotic analysis of these expressions leads to a rigorous derivation of universal hydrodynamic probability distribution functions. The latter lie in the KPZ universality class and are related to distributions from random matrix theory.

This is work in collaboration with Zeying Chen, Iori Hiki, William Mead, Masato Usui, Michael Wheeler and Tomohiro Sasamoto.

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Relevant papers: Phys. Rev. Lett. 120, 240601 (2018), A paper on 2-TASEP is in final stages of preparation

My aim is to review the connection between quantum-integrable long-range spin chains and Macdonald polynomials. I will introduce the main characters, the Haldane--Shastry spin chain and its q-deformation. Their remarkable properties include Yangian / quantum-loop symmetry at finite system size, and exact closed-form wave functions featuring (the zonal spherical case of) Jack / Macdonald polynomials. I will explain how the underlying algebraic structure, of affine Hecke algebras, can be used to construct a spin-version of Macdonald theory (spin Calogero--Sutherland / Ruijsenaars model) that reduces to the long-range spin chains in the 'freezing' limit.

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Relevant papers: BGHP hep-th/9301084, Uglov hep-th/9508145, JL 1801.05728, LPS 2004.13210

Non-symmetric(symmetric) Macdonald polynomials have been known to arise from many different aspects in mathematical physics, combinatorics and also probability theory i.e., the vertex model and the asymmetric simple exclusion process. In this talk, we are going to give an aspect of the non-symmetric Macdonald from the double affine Hecke algebra(DAHA) and how various formulas of the non-symmetric Macdonald polynomials are related i.e., the alcove walk formula(RY08) and the non-attacking filling formula(HHL07). This work is joint with Arun Ram.

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Relevant paper: Comparing formulas for type $\text{GL}_n$ Macdonald polynomials

The Kardar-Parisi-Zhang (KPZ) equation is the stochastic partial differential equation that models stochastic interface growth. I will present the construction of a stationary measure for the KPZ equation on a bounded interval with general inhomogeneous Neumann boundary conditions. This construction is a joint work with Ivan Corwin. In the talk I will mostly focus on the algebraic tools that are used in the study of stationary measures.

The favourite R-matrices and transfer matrices (which give the 6 vertex model) arise from the evaluation representation of the standard representation of the quantum group of sl(n). This is a level 0 representation of quantum affine sl(n) and the lattice models are based on tensor products of this representation. With Finn McGlade and Yaping Yang we have written a survey about the classification of these modules (by dominant weights for extremal weight modules and by Drinfeld polynomials for finite dimensional modules), their characters (which are q-Whittaker functions) and their crystals. I will try to sketch how I think this category (of level 0 modules) is the controlling structure for vertex operators, Fock spaces (Kyoto path model) and the Algebraic Bethe ansatz.

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We will explore Hamiltonian operators from the “symmetric function” perspective favored by Lam (arXiv:0507341). In the same way that Schur polynomials can be built up from power sum symmetric functions, we can produce other symmetric functions by replacing the power sums with other functions. The symmetric functions we obtain, which include Macdonald and LLT polynomials, will have “Schur-like” identities such as Cauchy and Pieri rules.

A question we will ask is “When can a lattice model partition function also be obtained from a Hamiltonian operator?” In the six-vertex case, the condition turns out to be that the Boltzmann weights are free fermionic. If we consider lattice models with charge, we obtain a similar condition that suggests that the weights of Brubaker, Bump, Buciumas, and Gustafsson (arXiv:1806.07776) may be (essentially) the only set of weights that corresponds to a Hamiltonian operator.

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Relevant papers: A combinatorial generalization of the Boson-Fermion correspondence

We describe a novel Yang-Baxter integrable vertex model, from which we construct a certain class of partition functions that are equal to the LLT polynomials of Lascoux, Leclerc, and Thibon. Using the vertex model formalism, we are able to prove many properties of these polynomials, including symmetry and a Cauchy identity. This is based on joint work with Sylvie Corteel, David Keating, and Jeremy Meza.

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Relevant papers: A vertex model for LLT polynomials, Colored Fermionic Vertex Models and Symmetric Functions, A Shuffle Theorem for Paths Under Any Line, A Combinatorial Formula for Macdonald Polynomials, Equivalences of LLT polynomials via lattice paths; Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties; Ribbon Tableaux and the Heisenberg Algebra, Ribbon lattices and ribbon function identities

I will describe how Whittaker functions related to both GLn(R) and SO2n+1(R) arise from a solvable random polymer model. The main tool towards this is A.N.Kirillov’s geometric lifting of the Robinson-Schensted-Knuth correspondence and certain variants of this, which include the geometric lifting of the Burge correspondence. A by-product of these studies is a combinatorial derivation of the Bump-Stade identities. In the core of this approach lies a volume preserving property of the geometric RSK and geometric Burge, which merits deeper investigation. This will be an overview of some works in collaboration with Bisi, O’Connell, Sepp ̈al ̈ainen.

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Relevant papers: Geometric RSK correspondence, Whittaker functions andsymmetrized random polymers; The geometric burge correspondence and the partition function of polymer replicas; Transition between characters of classical groups, decomposition of Gelfand-Tsetlin pattern and last passage percolation; Point-to-line polymers and orthogonal Whittaker functions; Introduction to tropical combinatorics; ArchimedeanL-factors on $GL(n) \times GL(n)$ and generalized Barnes integrals

Vertex operators originally arose in mathematical physics (string theory and soliton theory). They were applied to construct representations of affine Lie algebras by I. Frenkel and Kac, and an important algebraic fact emerged, the "boson-fermion correspondence". This concerns a Hamiltonian operator on the "fermionic Fock space". In some cases such a Hamiltonian can be related to the row transfer matrix for a solvable lattice model. The archetype for such a relation is Baxter's work relating the XYZ Hamiltonian with the 8-vertex model. A recent (2017) example is arXiv:1806.07776 where Brubaker, Buciumas, Bump and Gustafsson proved that the row transfer matrices of certain models motivated by the theory of metaplectic Whittaker functions could be realized by vertex operators on a version of the fermionic Fock space invented by Kashiwara, Miwa and Stern, that was previously applied by Lascoux, Leclerc and Thibon in the theory of ribbon symmetric functions.

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Relevant papers: Vertex operators, solvable lattice models and metaplectic Whittaker functions

Geometric constructions of quantum groups and their associated R-matrices arose in the early 90's and have been generalized further in recent works of Maulik, Okounkov, and others, creating a bridge between geometry and integrable lattice models. One nice aspect of this bridge is that the "hard" basis of one theory corresponds to the "easy" basis of the other.

We will analyze some of the lattice models we have already encountered in this seminar from this perspective. We first describe how both the torus fixed point basis and the basis of Schubert classes in the equivariant cohomology of the flag variety are manifest in the "Frozen Pipes" lattice model. This analysis is a straightforward generalization of results in a paper of Gorbunov, Korff, and Stroppel (see also the notes of Zinn-Justin) for the Grassmannian.

Then we describe how the fixed point basis and the basis of motivic Chern classes in the equivariant K-theory of the cotangent bundle of the flag variety appear (in a more novel way) in the Tokuyama model and colored Iwahori Whittaker model. Recent work of Aluffi, Mihalcea, Schürmann, and Su identifies these geometric bases with the Casselman and standard bases, respectively, of the Iwahori fixed vectors in the principal series representation, so this perspective allows us to make contact with formulas from p-adic representation theory, such as the Gindikin-Karpelevich formula and Bump-Nakasuji-Naruse conjecture. These ideas will be detailed in my forthcoming doctoral thesis.

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Relevant papers: Maulik and Okounkov, a more accessible summary of Maulik and Okounkov, Rimanyi (h-deformed Schubert calculus), Gorbunov, Korff, and Stroppel, Zinn-Justin's notes, Tokuyama model, Iwahori Whittaker model, Frozen Pipes, Motivic Chern classes and Iwahori invariants

Macdonald polynomials are a remarkable family of functions. They are a common generalization of many different families of special functions arising in the representation theory of reductive groups, including spherical functions and Whittaker functions.

In turn, Macdonald polynomials can be understood in terms of a certain representation of Cherednik's double affine Hecke algebra (DAHA), acting on polynomial functions on a torus.

Whittaker functions admit a natural generalization to the setting of metaplectic covers of reductive p-adic groups, which play a key role in the theory of Weyl group multiple Dirichlet series.

It turns out that Macdonald polynomials also admit a corresponding generalization, which can be understood in terms of a representation of the DAHA on the space of quasi-polynomial functions on a torus.

This is joint work with Jasper Stokman and Vidya Venkateswaran.

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Relevant papers: Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

Dual Grothendieck polynomials are the symmetric functions that are dual (under the Hall inner product, where the famous Schur polynomials give an orthonormal basis) to the symmetric Grothendieck polynomials, which are used to describe the K-theory ring of the Grassmannian. We can introduce extra parameters to obtain the refined dual Grothendieck polynomials. In this talk, we will translate the combinatorics of refined dual Grothendieck polynomials into the language of integrable lattice models. We then use lattice model techniques to derive a number of identities. We conclude by relating refined dual Grothendieck polynomials to TASEP through the last-passage percolation random matrices.

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Relevant papers: Refined dual Grothendieck polynomials, integrability, and the Schur measure

The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on the set of permutations, in which adjacent numbers i and j swap places at rate x_i - y_j if the larger number is clockwise of the smaller. Conjecturally, steady state probabilities can be written as a positive sum of (double) Schubert polynomials. We will start by giving some background on this model, including Cantini's result showing that the inhomogeneous TASEP is a solvable lattice model. We will then use his result to show that a large number of states -- those corresponding to the "evil-avoiding" permutations (permutations avoiding patterns 2413, 4132, 4213, 3214) -- have steady state probabilities which are proportional to a product of Schubert polynomials.

Based on joint work with Lauren Williams.

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I will talk about the stochastic six-vertex model from the integrable probability point of view. Namely, I will describe a new way to use the model's solvability by focusing on a specific recurrence relation (called local relation) on q-deformed moments of the height function. This relation leads to a short but non-constructive proof of the integral expressions for these moments. I will also sketch recent results about the colored stochastic six-vertex model, obtained using this technique.

Based on joint work with Alexey Bufetov.

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SSV polynomials are a new family of polynomials discovered by Sahi, Stokman, and Venkateswaran, generalizing both Macdonald polynomials and metaplectic Iwahori Whittaker functions. Similarly to both of these families, SSV polynomials satisfy a recursion coming from a Hecke algebra representation. We will use this recursion to give a combinatorial formula for SSV polynomials in terms of alcove walks, generalizing Ram and Yip's formula for Macdonald polynomials.

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Relevant papers: A combinatorial formula for SSV polynomials, Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials, A combinatorial formula for Macdonald polynomials, Metaplectic Iwahori Whittaker functions and supersymmetric lattice models

In this talk I will survey some applications of solvable lattice models to the study of special functions appearing in algebraic geometry, focusing on double Grothendieck polynomials. I will discuss some applications, like deriving combinatorial formulas for such functions. Time permitting, I will also mention structure coefficients.

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Relevant papers: ICERM slides on Schubert Calculus, Double Grothendieck polynomials and colored lattice models, Frozen Pipes: Lattice Models for Grothendieck Polynomials, Littlewood-Richardson coefficients for Grothendieck polynomials

The Izergin-Korepin analysis is originally a method to determine the exact forms of the domain wall boundary partition functions of the six-vertex model, which was originated in the works by Korepin and Izergin. In this talk, I will present the Izergin-Korepin analysis on the wavefunctions which are a larger class of partition functions. I will first explain the prototype case in detail, and then explain the case with reflecting boundary and show some applications.

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Relevant papers: Izergin-Korepin analysis, Quantum inverse scattering method, Metaplectic Whittaker Functions and Crystals of Type B

Values of matrix coefficients of p-adic groups can be written in terms of solvable lattice models. But the usual argument for that is ad hoc -- you first know the model and then show that the partition functions match the values of matrix coefficients. In my talk, I'll show how one can start with the p-adic side and naturally come up with solvable lattice models. Since these models are closely related to ones introduced in probability theory, I hope that it will give a new way to come up with models for integrable probability. My results extend Peter J. McNamara's approach from 0907.2675.

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Relevant papers: Metaplectic Whittaker Functions and Crystal Bases, Metaplectic Iwahori Whittaker Functions and Colored Crystal Bases (available by request: naprienko@stanford.edu)

Multi-species versions of several interacting particle systems, including ASEP, q-TAZRP, and k-exclusion processes, can be interpreted as random walks on Hecke algebras. In the talk I will discuss this connection and its probabilistic applications.

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Relevant papers: Interacting particle systems and random walks on Hecke algebras, Color-position symmetry in interacting particle systems

In this talk, we present a novel solvable lattice model which we term "stochastic symplectic ice" with stochastic weights and U-turn right boundary. The model can be interpreted probabilistically as a new interacting particle system in which particles jump alternately between right and left. We also introduce two colored versions of the model -- one of which involves "signed color" -- and the related stochastic dynamics. We then show how the functional equations and recursive relations for the partition functions of those models can be obtained using the Yang-Baxter equation. Finally, we show that the recursive relations satisfied by the partition function of one of the colored models are closely related to Demazure-Lusztig operators of type C.

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Relevant papers: Symplectic Ice, Stochastic symplectic ice

I will present a detailed description of a vertex model whose partition function computes all values of a basis of Iwahori Whittaker functions for unramified principal series of $GL_r(F)$ where $F$ is a non-archimedean field. In particular, there is a remarkable bijection between the data determining these values and the boundary data of the solvable lattice model. The vertex model can be described by colored paths in a grid and we will define its associated Boltzmann weights using a fusion process. We will show how its Yang-Baxter equation can be interpreted by Demazure-like operators on the representation theory side, and how the same model, but with different boundary conditions, also describes parahoric Whittaker functions.

Based on joint work with Ben Brubaker, Valentin Buciumas and Daniel Bump.

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Relevant papers: Colored five-vertex models and Demazure atoms, Colored Vertex Models and Iwahori Whittaker Functions, Metaplectic Iwahori Whittaker functions and supersymmetric lattice models

I will explain connections between stochastic particle systems (like q-TASEP or random polymers) and exactly solvable vertex models. More precisely, there is a whole family of results identifying random variables on the particle system side with certain quantities in a vertex model. There are several ways of establishing these results, including the Robinson-Schensted-Knuth correspondence. I will focus on bijectivisation of the Yang-Baxter equation for vertex models corresponding to q-Whittaker and the new spin q-Whittaker polynomials.

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We survey the ways in which solvability informs the study of special functions in algebra and representation theory. In particular, we explain how various natural Hecke algebra actions - from p-adic representation theory and from Schubert calculus - are manifested as Yang-Baxter equations of associated solvable lattice models. Then we describe how formal diagram identities of solvable models lead to new identities for the associated special functions.

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Relevant papers: Formal group laws, Algebraic Bethe Ansatz, Hecke algebras

This expository talk will concern various aspects of the stochastic six-vertex model. In particular, we will describe what sorts of asymptotic questions about this model integrable probabilists are interested in. Moreover, we will try to (at least partially) outline how its Yang-Baxter integrability can be useful in answering some of these questions.

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Relevant papers: Limit shapes and fluctuations, limit shape and local statistics, framework for evaluating q-moments

Solvable lattice models are statistical mechanical systems that can be studied exactly by a method of Baxter, based on the Yang-Baxter equation. This can be understood in terms of quantum groups. Recently particular examples showing symmetry with respect to the Lie super group $U_q(\widehat{\mathfrak{gl}}(r|n))$ arose in two very different contexts: the representation theory of $p$-adic groups, where such models were used by Brubaker, Buciumas, Bump and Gustafsson to study Iwahori Whittaker functions on metaplectic groups; and in integrable probability, in work recent of Aggarwal, Borodin and Wheeler. We will introduce the topic starting with Baxter's work on six vertex model, then look at more recent work explaining some of the ideas.

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Relevant papers: Tokuyama model for Spherical Whittaker Function, Colored Vertex Models and Iwahori Whittaker Functions, Metaplectic Iwahori Whittaker functions and Supersymmetric Lattice Models