Physics and Combinatorics

The following sections are included:

• PREFACE

• CONTENTS

In an earlier paper,¹³ we showed that the Hilbert scheme of points in the plane H_n = Hilbⁿ(ℂ²) can be identified with the Hilbert scheme of regular orbits ℂ²ⁿ //S_n. Using our earlier result and a recent result of Bridgeland, King and Reid,⁴ we prove vanishing theorems for tautological bundles on the Hilbert scheme. We apply the vanishing theorems to establish the conjectured character formula for diagonal harmonics of Garsia and the author.⁸ In particular we prove that the dimension of the space of diagonal harmonics is

We consider the statistical mechanics of quasi-particles which obey the exclusion statistics. Based on the recursion relation of the restricted partition function, we study the thermodynamic properties. We also apply this method to the sℓ_M,N supersymmetric system based on the super Yangian symmetry and the path configuration sum.

We describe our approach to the Forrester's conjectured constant term identity which is equivalent to a new kind of generalization of the Selberg integral. We also deal with the q–analogue of the conjecture in a parallel way and give a simple proof in the case N₁ = 2

The operator realizing a Dehn twist in quantum Teichmüller theory is diagonalized and continuous spectrum is obtained. This result is in agreement with the expected spectrum of conformal weights in quantum Liouville theory at c > 1. The completeness condition of the eigenvectors includes the integration measure which appeared in the representation theoretic approach to quantum Liouville theory by Ponsot and Teschner. The underlying quantum group structure is also revealed.

We construct birational representations of the symmetric, affine symmetric and extended affine symmetric groups on the affine spaces 𝔸ⁿ² and 𝔸^n(n+1)/2 which are tropical versions of the action of these groups on the set of transportation matrices and Lascoux–Schützenberger's action of the symmetric group on the set of semistandard Young tableaux. We show that tropical version of the Robinson–Schensted–Knuth correspondence (RSK for short) is a birational automorphism of the affine space 𝔸ⁿ² which intertwines two birational actions of the symmetric group on the space 𝔸ⁿ². We show that tropical version of RSK has many properties which are similar to those of the classical RSK. We study tropical version of statistics cocharge and proof, in particular, that it is invariant with respect to the birational action of the symmetric group on the space 𝔸^{n(n + 1)/2} constructed in this paper. We prove that tropical version of Schützenberger's involution satisfies the discrete Hirota–Miwa type equations (the Plücker relations for the 9-th variation of Schur functions). This result allows to find explicit formula for the tropical version of Schützenberger's involution as a generating function for certain weighted non–crossing lattice paths.

For any rational number p₀ ≥ 1 we prove an identity of Rogers–Ramanujany-Gordon–Andrews' type. Bijection between the space of states for XXZ model and that of XXX model is constructed.

Grothendieck polynomials represent Schubert varieties in the Grothendieck ring of the flag manifold (for Gl(n,ℂ)). We describe how general Grothendieck polynomials are related to those for Grasmann manifolds, which themselves are deformations of Schur functions.

The ninth variation of Schur functions associated with a general N × N matrix is studied. In particular, two types of tableau representations for them are constructed by using decomposition of matrices and properties of minor determinants.

Frenkel-Reshetikhin introduced q-characters of finite dimensional representations of quantum affine algebras [6]. We give a combinatorial algorithm to compute them for all simple modules. Our tool is t-analogue of the q-characters, which is similar to Kazhdan-Lusztig polynomials, and our algorithm has a resemblance with their definition. We need the theory of quiver varieties for the definition of t-analogues and the proof. But it appear only in the last section. The rest of the paper is devoted to an explanation of the algorithm, which one can read without the knowledge about quiver varieties. A proof is given only in part. A full proof will appear elsewhere.

We improve Barnes' proof of nonexistence of algebraic differential equations for his multiple gamma function. It is also proved that multiple q-gamma function satisfies no algebraic differential equation over a certain field. Furthermore, we consider an example of differential relations between several Barnes' multiple gamma functions in a special case.

Complete integrability of Calogero–Moser models is discussed from various angles. Calogero–Moser models and Toda models are best known examples of solvable many-particle dynamics on a line which are based on root systems. At the classical level, the former (C–M models) is integrable for elliptic potentials (Weierstraß ℘ function) and their various degenerate limits. The latter (Toda) has exponential potentials, which is obtained from the former as a special limit of the elliptic potential. Here we discuss quantum Calogero–Moser models based on any root system. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a Lax pair. (ii) Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalizations of the results known for the models based on the A series, i.e. su(N) type, root systems, which are treated rather explicitly in the Introduction. The independence of the conserved quantities (necessary for the Liouville integrability) obtained from the universal Lax pair is discussed in some detail.

Green functions of classical groups are determined by the data from Weyl groups and by certain combinatorial objects called symbols. Generalizing this, we define Green functions associated to complex reflection groups G(e, 1, n). We also construct Hall-Littlewood functions and Schur functions associated to G(e, 1, n), and show that the above Green functions are obtained as a transition matrix between those two symmetric functions. The details of this paper will appear elsewhere.

Thermodynamic Bethe ansatz equations for XXZ model at |Δ| ≥ 1 is simplified to an integral equation which has one unknown function. This equation is analytically continued to |Δ| < 1.

We study certain birational representation of symmetric group arising from inverse ultra discretization of the combinatorial R-matrix. This representation is related with the Weyl group action on the crystal bases and Lusztig's bijection on the canonical bases. We apply the result to the discrete Toda equation.