Bailey Lemma Research Articles

The Bailey Lemma and Kostka Polynomials

Using the theory of Kostka polynomials, we prove an A n−1 version of Bailey's lemma at integral level. Exploiting a new, conjectural expansion for Kostka numbers, this is then generalized to fractional levels, leading to a new expression for admissible characters of A(1) n−1 and to identities for A-type branching functions.

Extensions of the well-poised and elliptic well-poised Bailey lemma

We establish a number of extensions of the well-poised Bailey lemma and elliptic well-poised Bailey lemma. As application we prove some new transformation formulae for basic and elliptic hyper-geometric series, and embed some recent identities of Andrews, Berkovich and Spiridonov in a well-poised Bailey tree.

Partial Theta Functions. I. Beyond the Lost Notebook

It is shown how many of the partial theta function identities in Ramanujan's lost notebook can be generalized to infinite families of such identities. Key in our construction is the Bailey lemma and a new generalization of the Jacobi triple product identity. By computing residues around the poles of our identities we find a surprising connection between partial theta function identities and Garrett–Ismail–Stanton-type extensions of multisum Rogers–Ramanujan identities. 2000 Mathematics Subject Classification 05A30, 33D15, 33D90.

A PROBABILISTIC PROOF OF THE ROGERS–RAMANUJAN IDENTITIES

The asymptotic probability theory of conjugacy classes of the finite general groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains is given in this paper, leading to an elementary probabilistic proof of the Rogers–Ramanujan identities. This is compared with work on the uniform measure. The main case of Bailey's lemma is interpreted as finding eigenvectors of the transition matrix of a Markov chain. It is shown that the viewpoint of Markov chains extends to quivers.

Change of Base in Bailey Pairs

Versions of Bailey's lemma which change the base from q to q2 or q3 are given. Iterates of these versions give many new versions of multisum Rogers-Ramanujan identities.

An A₂ Bailey lemma and Rogers-Ramanujan-type identities

Using newqq-functions recently introduced by Hatayamaet al.and by (two of) the authors, we obtain an A2_2version of the classical Bailey lemma. We apply our result, which is distinct from the A2_2Bailey lemma of Milne and Lilly, to derive Rogers–Ramanujan-type identities for characters of the W3_3algebra.

Inhomogeneous Lattice Paths, Generalized Kostka Polynomials and A n −1 Supernomials

Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works and by Kashiwara's theory of crystal bases we define a statistic on paths yielding two novel classes of polynomials. One of these provides a generalization of the Kostka polynomials while the other, which we name the A$_{n-1}$ supernomial, is a $q$-deformation of the expansion coefficients of products of Schur polynomials. Many well-known results for Kostka polynomials are extended leading to representations of our polynomials in terms of a charge statistic on Littlewood-Richardson tableaux and in terms of fermionic configuration sums. Several identities for the generalized Kostka polynomials and the A$_{n-1}$ supernomials are proven or conjectured. Finally, a connection between the supernomials and Bailey's lemma is made.

The perturbations φ2,1 and φ1,5 of the minimal models M( p, p′) and the trinomial analogue of Bailey's lemma

We derive the fermionic polynomial generalizations of the characters of the integrable perturbations φ 2,1 and φ 1,5 of the general minimal M( p, p′) conformal field theory by use of the recently discovered trinomial analogue of Bailey's lemma. For φ 2,1 perturbations results are given for all models with 2 p > p′ and for φ 1,5 perturbations results for all models with p′/3 < p < p′/2 are obtained. For the φ 2,1 perturbation of the unitary case M( p, p + 1) we use the incidence matrix obtained from these character polynomials to discuss possible TBA equations. We also find that for φ 1,5 with 2 < p′/p < 5 2 and for φ 2,1 satisfying p < 2 p′ there are usually several different fermionic polynomials which lead to the identical bosonic polynomial. We interpret this to mean that in these cases the specification of the perturbing field is not sufficient to define the theory and that an independent statement of the choice of the proper vacuum must be made.

A Higher Level Bailey Lemma: Proof and Application

In a recent letter, new representations were proposed for the pair of sequences (γ,δ), as defined formally by Bailey in his famous lemma. Here we extend and prove this result, providing pairs (γ,δ) labelled by the Lie algebra AN − 1, two nonnegative integers l and k and a partition λ, whose parts do not exceed N − 1. Our results give rise to what we call a “higher level” Bailey lemma. As an application it is shown how this lemma can be applied to yield general q-series identities, which generalize some well-known results of Andrews and Bressoud.

Bailey flows and Bose-Fermi identities for the conformal coset models ( A1(1)) N × ( A1(1)) N′ /( A1(1)) N+ N′

We use the recently established higher-level Bailey lemma and Bose-Fermi polynomial identities for the minimal models M( p, p′) to demonstrate the existence of a Bailey flow from M( p, p′ ) to the coset models ( A 1 (1)) N × ( A 1 (1)) N′ /( A 1 (1)) N+ N′ where N is a positive integer and N′ is fractional, and to obtain Bose-Fetmi identities for these models. The fermionic side of these identities is expressed in terms of the fractional-level Cartan matrix introduced in the study of M( p, p′). Possible relations between Bailey and renormalization group flow are discussed.

Virasoro Character Identities from the Andrews–Bailey Construction

We prove q series identities between bosonic and fermionic representations of certain Virasoro characters. These identities include some of the conjectures made by the Stony Brook group as special cases. Our method is a direct application of Andrews' extensions of Bailey's lemma to recently obtained polynomial identities.

A Trinomial Analogue of Bailey's Lemma and N = 2 Superconformal Invariance

We propose and prove a trinomial version of the celebrated Bailey's lemma. As an application we obtain new fermionic representations for characters of some unitary as well as nonunitary models of N = 2 superconformal field theory (SCFT). We also establish interesting relations between N = 1 and N = 2 models of SCFT with central charges $(3/2)( 1 -{2(2 - 4\nu)^2}/{2(4\nu)})$ and $3(1 - 2/{4\nu})$. A number of new mock theta function identities are derived.

A Higher-Level Bailey Lemma

We propose a generalization of Bailey's lemma, useful for proving q-series identities. As an application, generalizations of Euler's identity, the Rogers–Ramanujan identities, and the Andrews–Gordon identities are derived. This generalized Bailey lemma also allows one to derive the branching functions of higher-level [Formula: see text] cosets.

Fermionic solution of the Andrews-Baxter-Forrester model. II. Proof of Melzer's polynomial identities

We compute the one-dimensional configuration sums of the ABF model using the fermionic technique introduced in part I of this paper. Combined with the results of Andrews, Baxter and Forrester, we find proof of polynomial identities for finitizations of the Virasoro characters $\chi_{b,a}^{(r-1,r)}(q)$ as conjectured by Melzer. In the thermodynamic limit these identities reproduce Rogers--Ramanujan type identities for the unitary minimal Virasoro characters, conjectured by the Stony Brook group. We also present a list of additional Virasoro character identities which follow from our proof of Melzer's identities and application of Bailey's lemma.

Consequences of the Al and Cl Bailey transform and Bailey lemma

In this paper we study some applications of the higher-dimensional generalization of the Bailey transform, Bailey lemma, and iterative ‘Bailey chain’ concept in the setting of basic hypergeometric series very well-poised on unitary A l , or symplectic C l , groups. The derivation of the C l , case is closely related to the previous analysis of the unitary A l , case. Let G denote A l , or C l . The G Bailey transform is obtained from a suitably modified G terminating very well-poised 4 φ 3 summation theorem and termwise transformations. It is then interpreted as a matrix inversion result for two infinite, lower-triangular matrices. This provides a higher-dimensional generalization of Andrews' matrix inversion formulation of the Bailey transform. As in the classical case, the concept of a G Bailey pair is introduced, and then inverted. This G inversion applied to the G terminating very well-poised 6 φ 5 summations yields G terminating balanced 3 φ 2 summations. The G Bailey lemma is obtained directly from a G terminating very well-poised 6 φ 5 summation theorem and the matrix inversion formulation of the G Bailey transform. It shows how to construct another G Bailey pair from an arbitrary G Bailey pair. The concepts of an ordinary G Bailey chain and a bilateral G Bailey chain are introduced. Finally, as an example, we give one A l , and one C l , q-Whipple transformation, and some of their applications. These include G, q-Dougall summations and G 4 φ 3 sears transformations. The classical case of all this work, corresponding to A l or equivalently U (2), contains an immense amount of the theory and application of one-variable basic hypergeometric series.

TheC ? Bailey transform and Bailey lemma

TheCl nonterminating Cl summation theorem is derived by appropriately specializing Gustafson's6ψ6 summation theorem for bilateral basic hypergeometric series very well-poised on symplecticCl groups. From this, the terminating6ϕ5 and, hence, terminating4ϕ3 summation theorem is obtained. A suitably modified4ϕ3 is then used to derive theCl generalization of the Bailey transform. The transform is then interpreted as a matrix inversion result for two infinite, lower-triangular matrices. This result is used to motivate the definition of theCl Bailey pair. TheCl generalization of Bailey's lemma is then proved. This result is inverted, and the concept of the bilateral Bailey chain is discussed. TheCl Bailey lemma is then used to obtain a connection coefficient result for generalCl littleq-Jacobi polynomials. All of this work is a natural extension of the unitaryAl, or equivalentlyU(l+1), case. The classical case, corresponding toA1 or equivalentlyU(2), contains an immense amount of the theory and application of one-variable basic hypergeometric series, including elegant proofs of the Rogers-Ramanujan-Schur identities. TheCl nonterminating6ϕ5 summation theorem is also used to recover C. Krattenthaler's multivariable summation which he utilized in deriving his refinement of the Bender-Knuth and MacMahon generating functions for certain sets of plane partitions.

Minima of higgs potentials corresponding to nonmaximal isotropy subgroups: M. Abud, G. Anastaze, P. Eckert, and H. Ruegg. Département de Physique Théorique, Université de Genève, 1211 Genève 4, Switzerland

We use the recently established higher-level Bailey lemma and Bose-Fermi polynomial identities for the minimal models M(p, p′) to demonstrate the existence of a Bailey flow from M(p, p′ ) to the coset models (A1(1))N × (A1(1))N′/(A1(1))N+N′ where N is a positive integer and N′ is fractional, and to obtain Bose-Fetmi identities for these models. The fermionic side of these identities is expressed in terms of the fractional-level Cartan matrix introduced in the study of M(p, p′). Possible relations between Bailey and renormalization group flow are discussed.