By using the Kang-Kashiwara-Misra-Miwa-Nakashima-Nakayashiki crystal base character formula for the basic $A_2^{(1)}$-module, and the principally specialized Weyl-Kac character formula, we obtain a Rogers-Ramanujan type combinatorial identity for colored partitions. The difference conditions between parts are given by the energy function of certain perfect $A_2^{(1)}$-crystal. We also recall some other identities for this type of colored partitions, but coming from the vertex operator constructions and with no apparent connection to the crystal base theory.
In their seminal work J. Lepowsky and R. L. Wilson gave a vertex-operator theoretic interpretation of Gordon-Andrews-Bressoud's generalization of Rogers-Ramanujan combinatorial identities, by constructing bases of vacuum spaces for the principal Heisenberg subalgebra of standard $\widehat{\mathfrak{sl}}_{2}$-modules, parametrized with partitions satisfying certain difference 2 conditions. In this paper we define quasi-particles in the principal picture of $\widehat{\mathfrak{sl}}_{2}$ and construct quasi-particle monomial bases of standard $\widehat{\mathfrak{sl}}_{2}$-modules for which princi…
A new type of polynomial analogue of the Rogers-Ramanujan identities is proven. Here the product-side of the Rogers-Ramanujan identities is replaced by a partial theta sum and the sum-side by a weighted sum over Schur polynomials.
We prove an identity for Hall--Littlewood symmetric functions labelled by the Lie algebra A_2. Through specialization this yields a simple proof of the A_2 Rogers--Ramanujan identities of Andrews, Schilling and the author.
Presented are polynomial identities which imply generalizations of Euler and Rogers--Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by establishing a graphical one-to-one correspondence between those two kinds of restricted partitions.
We explain the appearance of Rogers-Ramanujan series inside the tensor product of two basic $A_2^{(2)}$-modules, previously discovered by the first author in [F]. The key new ingredients are $(5,6)$ Virasoro minimal models and twisted modules for the Zamolodchikov $\WW_3$-algebra.
We refine and generalise a Rogers-Ramanujan type partition identity arising from crystal base theory. Our proof uses the variant of the method of weighted words recently introduced by the first author.
Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers-Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is representation theory of infinite dimensional Lie algebras, where various known interpretations of these identities have led to interesting applications. Motivated by their connections with Lie algebra representation theory, we give a new interpretation of a sum related to generalized Rogers-Ramanujan identities in terms of multi-color partitions.
Using new $q$-functions recently introduced by Hatayama et al. and by (two of) the authors, we obtain an A_2 version of the classical Bailey lemma. We apply our result, which is distinct from the A_2 Bailey lemma of Milne and Lilly, to derive Rogers-Ramanujan-type identities for characters of the W_3 algebra.
The Rogers-Ramanujan identities have been studied from the viewpoints of combinatorics, number theory, affine Lie algebras, statistical mechanics, and quantum field theory. This note connects the Rogers-Ramanujan identities with the finite general linear groups and the Hall-Littlewood polynomials of symmetric function theory.
We prove a q-series identity that generalises Macdonald's A_{2n}^{(2)} eta-function identity and the Rogers-Ramanujan identities. We conjecture our result to generalise even further to also include the Andrews-Gordon identities.
We consider expansions of products of theta-functions associated with arbitrary root systems in terms of nonsymmetric Macdonald polynomials at $t=\infty$ divided by their norms. The latter are identified with the graded characters of Demazure slices, some canonical quotients of thick (upper) level-one Demazure modules, directly related to recent theory of generalized (nonsymmetric) global Weyl modules. The symmetric Rogers-Ramanujan-type series considered by Cherednik-Feigin were expected to have some interpretation of this kind; the nonsymmetric setting appeared necessary to achieve this. As …
We highlight the role of q-series techniques in proving identities arising from knot theory. In particular, we prove Rogers-Ramanujan type identities for alternating knots as conjectured by Garoufalidis, Le and Zagier.
We use vertex operator algebras and intertwining operators to study certain substructures of standard $A_1^{(1)}$--modules, allowing us to conceptually obtain the classical Rogers--Ramanujan recursion. As a consequence we recover Feigin-Stoyanovsky's character formulas for the principal subspaces of the level 1 standard $A_1^{(1)}$--modules.
We conjecture polynomial identities which imply Rogers--Ramanujan type identities for branching functions associated with the cosets $({\cal G}^{(1)})_{\ell-1}\otimes ({\cal G}^{(1)})_{1} / ({\cal G}^{(1)})_{\ell}$, with ${\cal G}$=A$_{n-1}$ \mbox{$(\ell\geq 2)$}, D$_{n-1}$ $(\ell\geq 2)$, E$_{6,7,8}$ $(\ell=2)$. In support of our conjectures we establish the correct behaviour under level-rank duality for $\cal G$=A$_{n-1}$ and show that the A-D-E Rogers--Ramanujan identities have the expected $q\to 1^{-}$ asymptotics in terms of dilogarithm identities. Possible generalizations to arbitrary co…
The Rogers-Ramanujan identities and various analogous identities (Gordon, Andrews-Bressoud, Capparelli, etc.) form a family of very deep identities concerned with integer partitions. These identities (written in generating function form) are typically of the form "product side" equals "sum side," with the product side enumerating partitions obeying certain congruence conditions and the sum side obeying certain initial conditions and difference conditions (along with possibly other restrictions). We use symbolic computation to generate various such sum sides and then use Euler's algorithm to se…
In the theory of the Nil-DAHA Fourier transform, the inner products of q-Hermite polynomials for the measure function multiplied by a level one theta function are the key. They are used to obtain expansions of products of any number of such theta functions in terms of the q-Hermite polynomials. An ample family of modular functions satisfying Rogers-Ramanujan type identities for arbitrary (reduced, twisted) affine root systems is obtained as an application. A relation to Rogers dilogarithm and Nahm's conjecture is discussed. Some of our q-series can be identified with known ones, but their inte…
We utilize the technique of staircases and jagged partitions to provide analytic sum-sides to some old and new partition identities of Rogers-Ramanujan type. Firstly, we conjecture a class of new partition identities related to the principally specialized characters of certain level $2$ modules for the affine Lie algebra $A_9^{(2)}$. Secondly, we provide analytic sum-sides to some earlier conjectures of the authors. Next, we use these analytic sum-sides to discover a number of further generalizations. Lastly, we apply this technique to the well-known Capparelli identities and present analytic …
We present and prove Rogers-Schur-Ramanujan (Bose/Fermi) type identities for the Virasoro characters of the minimal model $M(p,p').$ The proof uses the continued fraction decomposition of $p'/p$ introduced by Takahashi and Suzuki for the study of the Bethe's Ansatz equations of the XXZ model and gives a general method to construct polynomial generalizations of the fermionic form of the characters which satisfy the same recursion relations as the bosonic polynomials of Forrester and Baxter. We use this method to get fermionic representations of the characters $χ_{r,s}^{(p,p')}$ for many classes…
Using the theory of intertwining operators for vertex operator algebras we show that the graded dimensions of the principal subspaces associated to the standard modules for $\hat{\goth{sl}(2)}$ satisfy certain classical recursion formulas of Rogers and Selberg. These recursions were exploited by Andrews in connection with Gordon's generalization of the Rogers--Ramanujan identities and with Andrews' related identities. The present work generalizes the authors' previous work on intertwining operators and the Rogers--Ramanujan recursion.
We generalize the "motivated proof" of the Rogers-Ramanujan identities given by Andrews and Baxter to provide an analogous "motivated proof" of Gordon's generalization of the Rogers-Ramanujan identities. Our main purpose is to provide insight into certain vertex-algebraic structure being developed.
