We introduce and study several combinatorial properties of a class of symmetric polynomials from the point of view of integrable vertex models in finite lattice. We introduce the $L$-operator related with the $U_q(sl_2)$ $R$-matrix, and construct the wavefunctions and their duals. We prove the exact correspondence between the wavefunctions and symmetric polynomials which is a quantum group deformation of the Grothendieck polynomials. This is proved by combining the matrix product method and an analysis on the domain wall boundary partition functions. As applications of the correspondence between the wavefunctions and symmetric polynomials, we derive several properties of the symmetric polynomials such as the determinant pairing formulas and the branching formulas by analyzing the domain wall boundary partition functions and the matrix elements of the $B$-operators.
We consider a homogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes. Our arguments are largely based on properties of a family of s…
Starting with any R-matrix with spectral parameter, obeying the Yang-Baxter equation and a unitarity condition, we construct the corresponding infinite dimensional quantum group U_{R} in term of a deformed oscillators algebra A_R. The realization we present is an infinite series, very similar to a vertex operator. Then, considering the integrable hierarchy naturally associated to A_{R}, we show that U_{R} provides its integrals of motion. The construction can be applied to any infinite dimensional quantum group, e.g. Yangians or elliptic quantum groups. Taking as an example the R-matrix of Y(N…
It is proved that the regularity of parafermion vertex operator algebras associated to integrable highest weight modules for affine Kac-Moody algebra A_1^{(1)} implies the C_2-cofiniteness of parafermion vertex operator algebras associated to integrable highest weight modules for any affine Kac-Moody algebra. In particular, the parafermion vertex operator algebra associated to an integrable highest weight module of small level for any affine Kac-Moody algebra is C_2-cofinite and has only finitely many irreducible modules. Also, the parafermion vertex operator algebras with level 1 are determin…
An $U_q(sl(n))$ invariant transfer matrix with periodic boundary conditions is analysed by means of the algebraic nested Bethe ansatz for the case of $q$ being a root of unity. The transfer matrix corresponds to a 2-dimensional vertex model on a torus with topological interaction w.r.t. the 3-dimensional interior of the torus. By means of finite size analysis we find the central charge of the corresponding Virasoro algebra as $c=(n-1) \left[1-n(n+1)/(r(r-1))\right] $.
We review and present new studies on the relation between the partition functions of integrable lattice models and symmetric polynomials, and its combinatorial representation theory based on the correspondence, including our work. In particular, we examine the correspondence between the wavefunctions of the XXZ type, Felderhof type and the boson type integrable models and symmetric polynomials such as the Schur, Grothendieck and symplectic Schur functions. We also give a brief report of our work on generalizing the correspondence between the Felderhof models and factorial Schur and symplectic …
Polyhedral realizations for crystal bases of integrable highest weight modules and combinatorial objects of type ${\rm A}^{(1)}_{n-1}$, ${\rm C}^{(1)}_{n-1}$, ${\rm A}^{(2)}_{2n-2}$, ${\rm D}^{(2)}_{n}$
A regular vertex operator algebra is a vertex operator algebra such that any weak module (without grading) is a direct sum of ordinary irreducible modules. In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular. We prove that the moonshine module vertex operator algebra $V^{\natural},$ the vertex operator algebras $L(l,0)$ associated with the integrable representations of affine algebras of level $l,$ the vertex operator algebras $L(c_{p,q},0)$ associated with irreducible highest weight representations for the discrete series of the Viraso…
We give new combinatorial formulas for decomposition of the tensor product of integrable highest weight modules over the classical Lie algebras of type $B, C, D$, and the branching decomposition of an integrable highest weight module with respect to a maximal Levi subalgebra of type $A$. This formula is based on a combinatorial model of classical crystals called spinor model. We show that our formulas extend in a bijective way various stable branching rules for classical groups to arbitrary highest weights, including the Littlewood restriction rules.
In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum $W_{1+\infty}$ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner - or vertex operator - of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of $\mathfrak{g…
The U_q(\hat{sl}_2) vertex model at q=0 with periodic boundary condition is an integrable cellular automaton in one-dimension. By the combinatorial Bethe ansatz, the initial value problem is solved for arbitrary states in terms of an ultradiscrete analogue of the Riemann theta function with rational characteristics.
We examine and present new combinatorics for the Schur polynomials from the viewpoint of quantum integrability. We introduce and analyze an integrable six-vertex model which can be viewed as a certain degeneration model from a t-deformed boson model. By a detailed analysis of the wavefunction from the quantum inverse scattering method, we present a novel combinatorial formula which expresses the Schur polynomials by using an additional parameter, which is in the same sense but different from the Tokuyama formula. We also give an algebraic analytic proof for the Cauchy identity and make applica…
The work is devoted to a probably new connection between deformed Virasoro algebra and quantum $\widehat{\mathfrak{sl}}_2$. We give an explicit realization of Virasoro current via vertex operators of level 1 integrable representation of $\widehat{\mathfrak{sl}}_2$. The same is done for a twisted version of deformed Virasoro algebra.
The structure of the parafermion vertex operator algebra associated to an integrable highest weight module for any affine Kac-Moody algebra is studied. In particular, a set of generators for this algebra has been determined.
We show that a set of local admissible fields generates a vertex algebra. For an affine Lie algebra $\tilde\goth g$ we construct the corresponding level $k$ vertex operator algebra and we show that level $k$ highest weight $\tilde\goth g$-modules are modules for this vertex operator algebra. We determine the set of annihilating fields of level $k$ standard modules and we study the corresponding loop $\tilde\goth g$ module---the set of relations that defines standard modules. In the case when $\tilde\goth g$ is of type $A_1^{(1)}$, we construct bases of standard modules parameterized by colored…
A module M over a vertex algebra V is half-integrable if a_n act locally nilpotently on M for all a in V, m in M, n>0. We study half-integrable modules over sheaves of twisted chiral differential operators (TCDO) on a smooth variety X. We prove an equivalence between certain categories of half-integrable modules over TCDO and categories of (twisted) D-modules on X.
In this paper, the structure of the parafermion vertex operator algebra associated to an integrable highest weight module for simple affine Lie superalgebra $osp(1|2n)$ is studied. Particularly, we determine the generators for this algebra.
In this paper we construct bases of standard (i.e. integrable highest weight) modules $L(Λ)$ for affine Lie algebra of type $B_2\sp{(1)}$ consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces $W(Λ)$ of $L(Λ)$ by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence $W(kΛ_0)$ for $B_2\sp{(1)}$ and the integrable highest weight module $L(kΛ_0)$ for $A_1\sp{(1)}$ have the same parametrization of combinatorial bases and the same presentation $\mathcal P/\mathcal I$\…
Haisheng Li showed that given a module (W,Y_W(\cdot,x)) for a vertex algebra (V,Y(\cdot,x)), one can obtain a new V-module W^Δ = (W,Y_W(Δ(x)\cdot,x)) if Δ(x) satisfies certain natural conditions. Li presented a collection of such Δ-operators for V=L(k,0) (a vertex operator algebra associated with an affine Lie algebras, k a positive integer). In this paper, for each irreducible L(k,0)-module W, we find a highest weight vector of W^Δ when Δis associated with a miniscule coweight. From this we completely determine the action of these Δ-operators on the set of isomorphism equivalence classes of L…
We introduce generalized energies for a class of U_q(D^{(1)}_n) crystals by using the piecewise linear functions that are building blocks of the combinatorial R. They include the conventional energy in the theory of affine crystals as a special case. It is shown that the generalized energies count the particles and anti-particles in a quadrant of the two dimensional lattice generated by time evolutions of an integrable D^{(1)}_n cellular automaton. Explicit formulas are conjectured for some of them in the form of ultradiscrete tau functions.
There are many analogies between codes, lattices, and vertex operator algebras. For example, extremal objects are good examples of combinatorial, spherical, and conformal designs. In this study, we investigated these objects from the aspect of design theory.
We construct embeddings of $\widehat{\mathfrak{sl}}_2$ in lattice vertex algebras by composing the Wakimoto realization with the Friedan-Martinec-Shenker bosonization. The Kac-Wakimoto hierarchy then gives rise to two new hierarchies of integrable, non-autonomous, non-linear partial differential equations. A new feature of our construction is that it works for any value of the central element of $\widehat{\mathfrak{sl}}_2$; that is, the level becomes a parameter in the equations.
We consider two different methods of associating vertex algebraic structures with the level $1$ principal subspaces for $U_q (\widehat{\mathfrak{sl}}_2)$. In the first approach, we introduce certain commutative operators and study the corresponding vertex algebra and its module. We find combinatorial bases for these objects and show that they coincide with the principal subspace bases found by B. L. Feigin and A. V. Stoyanovsky. In the second approach, we introduce the, so-called nonlocal $\underline{\mathsf{q}}$-vertex algebras, investigate their properties and construct the nonlocal $\underl…
The spectral decomposition of the path space of the vertex model associated to the vector representation of the quantized affine algebra $U_q(\hat{sl}_n)$ is studied. We give a one-to-one correspondence between the spin configurations and the semi-standard tableaux of skew Young diagrams. As a result we obtain a formula of the characters for the degeneracy of the spectrum in terms of skew Schur functions. We conjecture that our result describes the $sl_n$-module contents of the Yangian $Y(sl_n)$-module structures of the level 1 integrable modules of the affine Lie algebra $\hat{sl}_n$. An anal…
We consider global generation of sheaves of coinvariants on the moduli space of curves given by simple modules over certain vertex operator algebras, extending results for affine VOAs at integrable levels on stable pointed rational curves. Examples where global generation fails, and further evidence of positivity are given.
We study graded nonlocal $\underline{\mathsf{q}}$-vertex algebras and we prove that they can be generated by certain sets of vertex operators. As an application, we consider the family of graded nonlocal $\underline{\mathsf{q}}$-vertex algebras $V_{c,1}$, $c\geq 1$, associated with the principal subspaces $W(cΛ_0)$ of the integrable highest weight $U_q (\hat{\mathfrak{sl}}_2)$-modules $L(cΛ_0)$. Using quantum integrability, we derive combinatorial bases for $V_{c,1}$ and compute the corresponding character formulae.
Vertex operators associated with level two $U_q(\widehat{sl}_2)$ modules are constructed explicitly using bosons and fermions. An integral formula is derived for the trace of products of vertex operators. These results are applied to give $n$-point spin correlation functions of an integrable $S=1$ quantum spin chain, extending an earlier work of Jimbo {\em et al} for the case $S=1/2$.
Let Uq(g) be the quantum affine superalgebra associated with an affine Kac-Moody superalgebra g which belongs to the three series osp(1|2n)^(1),sl(1|2n)^(2) and osp(2|2n)^(2). We develop vertex operator constructions for the level 1 irreducible integrable highest weight representations and classify the finite dimensional irreducible representations of Uq(g). This makes essential use of the Drinfeld realisation for Uq(g), and quantum correspondences between affine Kac-Moody superalgebras, developed in earlier papers.
We calculate chiral rings of the N=2 vertex algebras constructed from the combinatorial data of toric mirror symmetry and show that they coincide with the description of stringy cohomology conjectured previously in a joint work with A. Mavlyutov.
