arXiv subjects

R. Kedem

Publications and source records attributed to R. Kedem.

At least 19 recordsLinked to original sources

arXiv2014

A path model for Whittaker vectors

In this paper we construct weighted path models to compute Whittaker vectors in the completion of Verma modules, as well as Whittaker functions of fundamental type, for all finite-dimensional simple Lie algebras, affine Lie algebras, and the quantum algebra $U_q(\mathfrak{sl}_{r+1})$. This leads to series expressions for the Whittaker functions. We show how this construction leads directly to the quantum Toda equations satisfied by these functions, and to the $q$-difference equations in the quantum case. We investigate the critical limit of affine Whittaker functions computed in this way.

math.RT↗

arXiv2010

Q-systems, Heaps, Paths and Cluster Positivity

We consider the cluster algebra associated to the $Q$-system for $A_r$ as a tool for relating $Q$-system solutions to all possible sets of initial data. We show that the conserved quantities of the $Q$-system are partition functions for hard particles on particular target graphs with weights, which are determined by the choice of initial data. This allows us to interpret the simplest solutions of the Q-system as generating functions for Viennot's heaps on these target graphs, and equivalently as generating functions of weighted paths on suitable dual target graphs. The generating functions take the form of finite continued fractions. In this setting, the cluster mutations correspond to local rearrangements of the fractions which leave their final value unchanged. Finally, the general solutions of the $Q$-system are interpreted as partition functions for strongly non-intersecting families of lattice paths on target lattices. This expresses all cluster variables as manifestly positive Laurent polynomials of any initial data, thus proving the cluster positivity conjecture for the $A_r$ $Q$-system. We also give an alternative formulation in terms of domino tilings of deformed Aztec diamonds with defects.

math.CO↗

arXiv2009

Discrete non-commutative integrability: the proof of a conjecture by M. Kontsevich

We prove a conjecture of Kontsevich regarding the solutions of rank two recursion relations for non-commutative variables which, in the commutative case, reduce to rank two cluster algebras of affine type. The conjecture states that solutions are positive Laurent polynomials in the initial cluster variables. We prove this by use of a non-commutative version of the path models which we used for the commutative case.

math-ph↗

arXiv2009

Positivity of the T-system cluster algebra

We give the path model solution for the cluster algebra variables of the $A_r$ $T$-system with generic boundary conditions. The solutions are partition functions of (strongly) non-intersecting paths on weighted graphs. The graphs are the same as those constructed for the $Q$-system in our earlier work, and depend on the seed or initial data in terms of which the solutions are given. The weights are "time-dependent" where "time" is the extra parameter which distinguishes the $T$-system from the $Q$-system, usually identified as the spectral parameter in the context of representation theory. The path model is alternatively described on a graph with non-commutative weights, and cluster mutations are interpreted as non-commutative continued fraction rearrangements. As a consequence, the solution is a positive Laurent polynomial of the seed data.

math.CO↗

arXiv2008

Proof of the combinatorial Kirillov-Reshetikhin conjecture

In this paper we give a direct proof of the equality of certain generating function associated with tensor product multiplicities of Kirillov-Reshetikhin modules for each simple Lie algebra g. Together with the theorems of Nakajima and Hernandez, this gives the proof of the combinatorial version of the Kirillov-Reshetikhin conjecture, which gives tensor product multiplicities in terms of restricted fermionic summations.

math.QA↗

arXiv2002

Spaces of coinvariants and fusion product I. From equivalence theorem to Kostka polynomials

The fusion rule gives the dimensions of spaces of conformal blocks in the WZW theory. We prove a dimension formula similar to the fusion rulefor spaces of coinvariants of affine Lie algebras g^. An equivalence of filtered spaces is established between spaces of coinvariants of two objects: highest weight g^-modules and tensor products of finite-dimensional evaluation representations of g\otimes\C[t]. In the sl_2 case we prove that their associated graded spaces are isomorphic to the spaces of coinvariants of fusion products, and that their Hilbert polynomials are the level-restricted Kostka polynomials.

math.QA↗

arXiv2002

Spaces of coinvariants and fusion product II. Affine sl_2 character formulas in terms of Kostka polynomials

In this paper, we continue our study of the Hilbert polynomials of coinvariants begun in our previous work math.QA/0205324 (paper I). We describe the sl_n-fusion products for symmetric tensor representations following the method of Feigin and Feigin, and show that their Hilbert polynomials are A_{n-1}-supernomials. We identify the fusion product of arbitrary irreducible sl_n-modules with the fusion product of their resctriction to sl_{n-1}. Then using the equivalence theorem from paper I and the results above for sl_3, we give a fermionic formula for the Hilbert polynomials of a class of affine sl_2-coinvariants in terms of the level-restricted Kostka polynomials. The coinvariants under consideration are a generalization of the coinvariants studied in [FKLMM]. Our formula differs from the fermionic formula established in [FKLMM] and implies the alternating sum formula conjectured in [FL] for this case.

math.QA↗

arXiv2001

Combinatorics of the $\hat{sl}_2$ Spaces of Coinvariants III

We give the fermionic character formulas for the spaces of coinvariants obtained from level $k$ integrable representations of $\hat{\mathfrak sl}_2$. We establish the functional realization of the spaces dual to the coinvariant spaces. We parameterize functions in the dual spaces by rigged partitions, and prove the recursion relations for the sets of rigged partitions.

math.QA↗

arXiv2001

Combinatorics of the $\hat{sl}_2$ Spaces of Coinvariants

We consider two types of quotients of the integrable modules of $\hat{sl}_2$. These spaces of coinvariants have dimensions described in terms of the Verlinde algebra of level-$k$. We describe monomial bases for the spaces of coinvariants, which leads to a fermionic description of these spaces. For $k=1$, we give the explicit formulas for the characters. We also present recursion relations satisfied by the characters and the monomial bases.

math-ph↗

arXiv2001

Combinatorics of the $\hat{sl_2}$ spaces of coinvariants II

The spaces of coinvariants are quotient spaces of integrable $\hat{sl_2}$ modules by subspaces generated by actions of certain subalgebras labeled by a set of points on a complex line. When all the points are distinct, the spaces of coinvariants essentially coincide with the spaces of conformal blocks in the WZW conformal field theory and their dimensions are given by the Verlinde rule. We describe monomial bases for the $\wsl$ spaces of coinvariants, In particular, we prove that the spaces of coinvariants have the same dimensions when all the points coincide. We establish recursive relations satisfied by the monomial bases and the corresponding characters of the spaces of coinvariants. For the proof we use filtrations of the $\hat{\mathfrak{sl}}_2$ modules, and further filtrations on the adjoint graded spaces for the first filtrations. This paper is the continuation of [FKLMM].

math.QA↗

arXiv1995

New Level-0 Action of $U_q(\widehat{sl}_2)$ on Level-1 Modules

A level-0 action of $U_q(\widehat{sl}_2)$ is defined on the sum of level-1 irreducible highest weight modules. With the aid of the affine Hecke algebras, this action is realized on the basis created by the vertex operators. This is a $q$-analogue of the Yangian symmetry in conformal field theory.

q-alg↗

arXiv1995

Level-0 structure of level-1 $U_q(\widehat{sl}_2)$-modules and Macdonald polynomials

The level-$1$ integrable highest weight modules of $U_q(\widehat{sl}_2)$ admit a level-$0$ action of the same algebra. This action is defined using the affine Hecke algebra and the basis of the level-$1$ module generated by components of vertex operators. Each level-$1$ module is a direct sum of finite-dimensional irreducible level-$0$ modules, whose highest weight vector is expressed in terms of Macdonald polynomials. This decomposition leads to the fermionic character formula for the level-$1$ modules.

q-alg↗

arXiv1994

XXZ chain with a boundary

The $\XXZ$ spin chain with a boundary magnetic field $h$ is considered, using the vertex operator approach to diagonalize the Hamiltonian. We find explicit bosonic formulas for the two vacuum vectors with zero particle content. There are three distinct regions when $h\geq0$, in which the structure of the vacuum states is different. Excited states are given by the action of vertex operators on the vacuum states. We derive the boundary $S$-matrix and present an integral formula for the correlation functions. The boundary magnetization exhibits boundary hysteresis. We also discuss the rational limit, the $\XXX$ model.

hep-th↗

arXiv1994

An elliptic quantum algebra for $\widehat{sl}_2$

An elliptic deformation of $\widehat{sl}_2$ is proposed. Our presentation of the algebra is based on the relation $RLL=LLR^*$, where $R$ and $R^*$ are eight-vertex $R$-matrices with the elliptic moduli chosen differently. In the trigonometric limit, this algebra reduces to a quotient of that proposed by Reshetikhin and Semenov-Tian-Shansky. Conjectures concerning highest weight modules and vertex operators are formulated, and the physical interpretation of $R^*$ is discussed.

hep-th↗

arXiv1994

Notes on highest weight modules of the elliptic algebra ${\cal A}_{q,p}\left(\widehat{sl}_2\right)$

We discuss a construction of highest weight modules for the recently defined elliptic algebra ${\cal A}_{q,p}(\widehat{sl}_2)$, and make several conjectures concerning them. The modules are generated by the action of the components of the operator $L$ on the highest weight vectors. We introduce the vertex operators $Φ$ and $Ψ^*$ through their commutation relations with the $L$-operator. We present ordering rules for the $L$- and $Φ$-operators and find an upper bound for the number of linearly independent vectors generated by them, which agrees with the known characters of $\widehat{sl}_2$-modules.

hep-th↗

arXiv1993

Virasoro Characters from Bethe Equations for the Critical Ferromagnetic Three-State Potts Model

We obtain new fermionic sum representations for the Virasoro characters of the confromal field theory describing the ferromagnetic three-state Potts spin chain. These arise from the fermionic quasi-particle excitations derived from the Bethe equations for the eigenvalues of the hamiltonian. In the conformal scaling limit, the Bethe equations provide a description of the spectrum in terms of one genuine quasi-particle, and two ``ghost'' excitations with a limited microscopic momentum range. This description is reflected in the structure of the character formulas, and suggests a connection with the integrable perturbation of dimensions (2/3,2/3)$^+$ which breaks the $S_3$ symmetry of the conformal field theory down to $Z_2$.

hep-th↗

arXiv1993

Fermionic Sum Representations for Conformal Field Theory Characters

We present sum representations for all characters of the unitary Virasoro minimal models. They can be viewed as fermionic companions of the Rocha-Caridi sum representations, the latter related to the (bosonic) Feigin-Fuchs-Felder construction. We also give fermionic representations for certain characters of the general $(G^{(1)})_k \times (G^{(1)})_l \over (G^{(1)})_{k+l}}$ coset conformal field theories, the non-unitary minimal models ${\cal M}(p,p+2)$ and ${\cal M}(p,kp+1)$, the $N$=2 superconformal series, and the $\ZZ_N$-parafermion theories, and relate the $q\to 1$ behaviour of all these fermionic sum representations to the thermodynamic Bethe Ansatz.

hep-th↗

arXiv1993

Quasi-Particles, Conformal Field Theory, and $q$-Series

We review recent results concerning the representation of conformal field theory characters in terms of fermionic quasi-particle excitations, and describe in detail their construction in the case of the integrable three-state Potts chain. These fermionic representations are $q$-series which are generalizations of the sums occurring in the Rogers-Ramanujan identities. (To appear in the proceedings of ``Yang-Baxter Equations in Paris'', July 1992, J.-M.~Maillard (ed.).)

hep-th↗