This work is dedicated to sl n+1 -related integrable stochastic vertex models; we call such models coloured. We prove several results about these models, which include the following:(1) We construct the basis of (rational) eigenfunctions of the coloured transfer-matrices as partition functions of our lattice models with certain boundary conditions. Similarly, we construct a dual basis and prove the corresponding orthogonality relations and Plancherel formulae.(2) We derive a variety of combinatorial properties of those eigenfunctions, such as branching rules, exchange relations under Hecke divided-difference operators, (skew) Cauchy identities of different types, and monomial expansions.(3) We show that our eigenfunctions are certain (non-obvious) reductions of the nested Bethe Ansatz eigenfunctions.(5) We demonstrate how the coloured-uncoloured match degenerates to the coloured (or multi-species) versions of the ASEP, q-PushTASEP, and the q-boson model.(6) We show how our eigenfunctions relate to non-symmetric Cherednik-Macdonald theory, and we make use of this connection to prove a probabilistic matching result by applying Cherednik-Dunkl operators to the corresponding non-symmetric Cauchy identity. ContentsChapter 1. Introduction 1.1. Preface 1.2. The model 1.3. The transfer-matrix and its eigenfunctions 1.4. Plancherel theory 1.5. Summation identities, recursive relations, monomial expansions 1.6. Matching distributions 1.7. Matching for interacting particle systems 1.8. Asymptotics 1.9. A word about extensions 1.10. Acknowledgments Chapter 2. Rank-n vertex models 2.1. Stochastic U q ( sl n+1 ) R-matrix 2.2. Higher-spin L and M -matrices 2.3. Intertwining equations 2.4. Colour-blindness results 2.5. Stochastic weights Chapter 3. Row operators and non-symmetric rational functions 3.1. Space of states and row operators 3.2. Commutation relations 3.3. Coloured compositions 3.4. The rational non-symmetric functions f µ and g µ 3.5. Permuted boundary conditions 3.6. Pre-fused functions Chapter 4. Branching rules and summation identities 4.1. Skew functions f µ/ν and g µ/ν 4.2. Branching rules 4.3. Summation identities of Mimachi-Noumi type 4.4. Symmetric rational function G µ/ν 4.5. Cauchy identities Chapter 5. Recursive properties and symmetries 5.1. Factorization of f δ 5.2. Hecke algebra and its polynomial representation 5.3. Exchange relations for f µ 5.4. Symmetry in (x i , x i+1) for µ i = µ i+1 5.5. Relationship between f µ and f σ δ 5.6. Relationship between f µ and g µ5.7. Relationship between f σ δ and g µ 5.8. Exchange relations for g µ 5.9. Reduction to non-symmetric Hall-Littlewood polynomials 5.10. Eigenrelation for the non-symmetric Hall-Littlewood polynomials Chapter 6. Monomial expansions: permutation graphs 6.1. Warm-up: F -matrices for two-site spin chains 6.2. N -site R-matrices 6.3. Permutation graphs 6.4. Reversed permutation graphs 6.5. F -matrices for spin chains of generic length 6.6. Column operators 6.7. Monomial expansions Chapter 7. Monomial expansions: degenerations of nested Bethe ve...

Abstract. We derive a matrix product formula for symmetric Macdonald polynomials.Our results are obtained by constructing polynomial solutions of deformed Knizhnik-Zamolodchikov equations, which arise by considering representations of the Zamolodchikov-Faddeev and Yang-Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalised probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalisations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1.

In Reconstructing the Cognitive World, Michael Wheeler argues that we should turn away from the generically Cartesian philosophical foundations of much contemporary cognitive science research and proposes instead a Heideggerian approach. Wheeler begins with an interpretation of Descartes. He defines Cartesian psychology as a conceptual framework of explanatory principles and shows how each of these principles is part of the deep assumptions of orthodox cognitive science (both classical and connectionist). Wheeler then turns to Heidegger's radically non-Cartesian account of everyday cognition, which, he argues, can be used to articulate the philosophical foundations of a genuinely non-Cartesian cognitive science. Finding that Heidegger's critique of Cartesian thinking falls short, even when supported by Hubert Dreyfus's influential critique of orthodox artificial intelligence, Wheeler suggests a new Heideggerian approach. He points to recent research in "embodied-embedded" cognitive science and proposes a Heideggerian framework to identify, amplify, and clarify the underlying philosophical foundations of this new work. He focuses much of his investigation on recent work in artificial intelligence-oriented robotics, discussing, among other topics, the nature and status of representational explanation, and whether (and to what extent) cognition is computation rather than a noncomputational phenomenon best described in the language of dynamical systems theory. Wheeler's argument draws on analytic philosophy, continental philosophy, and empirical work to "reconstruct" the philosophical foundations of cognitive science in a time of a fundamental shift away from a generically Cartesian approach. His analysis demonstrates that Heideggerian continental philosophy and naturalistic cognitive science need not be mutually exclusive and shows further that a Heideggerian framework can act as the "conceptual glue" for new work in cognitive science. Bradford Books imprint

We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary condition, the height function at the origin fluctuates asymptotically (in large time τ ) according to the Tracy-Widom GOE distribution on the τ 1/3 scale. This is the first example of KPZ asymptotics for a half-space system outside the class of free-fermionic/determinantal/Pfaffian models.Our main tool in this analysis is a new class of probability measures on Young diagrams that we call half-space Macdonald processes, as well as two surprising relations. The first relates a special (Hall-Littlewood) case of these measures to the half-space stochastic six-vertex model (which further limits to ASEP) using a Yang-Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogs via a refined Littlewood identity.study the statistics of the number of particles in the system when started empty, and with boundary conditions tuned to this critical point. We prove that after a very long time τ , the random variable scales (around its law of large numbers centering) like τ 1/3 and converges in this scale weakly to the GOE Tracy-Widom distribution (Theorem A). Further, our results also shed light on the distribution of the solution to the KPZ equation with Neumann boundary condition (Theorem B), which arises as a limit of the height function of weakly asymmetric half-line ASEP around this critical point [CS16]. This is the first proof of (KPZ / random-matrix-theoretic) asymptotics in a non free-fermionic half-space model. Free-fermionic full-space systems have been well-studied via robust mathematical approaches, in particular the Schur processes [OR03]. These are determinantal systems, meaning that correlation functions are written as determinants of a large matrix. The half-space analog of such systems are Pfaffian Schur processes [Rai00, BR01a, BR05, SI04], whose correlation functions are given via Pfaffians. The full and half-space TASEP (where jumps only go in one direction) and a small handful of other models fit into the free-fermionic framework.ASEP and many other important models do not fit into the free-fermionic framework. In the last decade, starting from the work of Tracy and Widom on ASEP (on the full line) [TW09], many KPZ type limit theorems have been obtained for non-free fermionic models in a full-space. These results have helped refine and expand the notion of KPZ universality. Some attempts have been made to study similar half-space systems, but until now no method has yielded rigorous distributional asymptotics without a Pfaffian structure. Among the existing works on non freefermionic half-space systems, [OSZ14] studied the Log-Gamma directed polymer in a half-quadrant using properties of the geometric RSK algorithm on symmetric matrices, and conjectured integral formulas, but these are presently not amenable for asymptotic analysis. Using...

Abstract. We prove and conjecture some new symmetric function identities, which equate the generating series of 1. Plane partitions, subject to certain restrictions and weightings, and 2. Alternating sign matrices, subject to certain symmetry properties. The left hand side of each of our identities is a simple refinement of a relevant Cauchy or Littlewood identity, allowing them to be interpreted as generating series for plane partitions. The right hand side of each identity is a partition function of the six-vertex model, on a relevant domain. These can be interpreted as generating series for alternating sign matrices, using the well known bijection with six-vertex model configurations.