The Rogers-Selberg recursions, the Gordon-Andrews identities and intertwining operators

IdentityFinder and some new identities of Rogers-Ramanujan type

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…

On certain identities involving Nahm-type sums with double poles

We prove certain Nahm-type sum representations for the (odd modulus) Andrews-Gordon identities, the (even modulus) Andrews-Bressoud identities, and Rogers' false theta functions. These identities are motivated on one hand by a recent work of C. Jennings-Shaffer and one of us on double pole series, and, on the other hand, by Córdova, Gaiotto and Shao's work on defect Schur's indices.

Quasi-particles in the principal picture of $\widehat{\mathfrak{sl}}_{2}$ and Rogers-Ramanujan-type identities

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…

关于三重$q$-级数与Rogers-Hecke型级数的一个变换

On a Transformation of Triple $q$-Series and Rogers-Hecke Type Series

本研究利用$q$-指数微分算子推广Sears $_4ϕ_3$变换公式，结合解析函数原点$q$-级数展开得到三重$q$-级数变换公式，并将其应用于Rogers-Hecke型级数，获得100多个相关恒等式。

Bethe's states for generalized XXX and XXZ models

For any rational number $p_0\ge 1$ we prove an identity of Rogers-Ramanujan-Gordon-Andrews' type. Bijection between the space of states for XXZ model and that of XXX model is constructed.

Intertwining vertex operators and certain representations of sl(n)^

We study the principal subspaces, introduced by B. Feigin and A. Stoyanovsky, of the level 1 standard modules for $\hat{\goth{sl}(l+1)}$ with $l \geq 2$. In this paper we construct exact sequences which give us a complete set of recursions that characterize the graded dimensions of the principal subspaces of these representations. This problem can be viewed as a continuation of a new program to obtain Rogers-Ramanujan-type recursions, which was initiated by S. Capparelli, J. Lepowsky and A. Milas. In order to prove the exactness of the sequences we use intertwining vertex operators and we supp…

边界最小模型与罗杰斯-拉马努金恒等式

Boundary minimal models and the Rogers-Ramanujan identities

研究确定了Virasoro顶点代数的不可约模在经典自由条件下的情况，发现仅在边界最小模型中成立，并利用安德鲁斯-戈登推广的罗杰斯-拉马努金恒等式进行证明。

Ghost series and a motivated proof of the Andrews-Bressoud identities

We present what we call a "motivated proof" of the Andrews-Bressoud partition identities for even moduli. A "motivated proof" of the Rogers-Ramanujan identities was given by G. E. Andrews and R. J. Baxter, and this proof was generalized to the odd-moduli case of Gordon's identities by J. Lepowsky and M. Zhu. Recently, a "motivated proof" of the somewhat analogous Göllnitz-Gordon-Andrews identities has been found. In the present work, we introduce "shelves" of formal series incorporating what we call "ghost series," which allow us to pass from one shelf to the next via natural recursions, leadi…

A proof of conjectured partition identities of Nandi

We generalize the theory of linked partition ideals due to Andrews using finite automata in formal language theory and apply it to prove three Rogers--Ramanujan type identities of modulo 14 that were posed by Nandi through vertex operator theoretic construction of the level 4 standard modules of the affine Lie algebra $A^{(2)}_{2}$.

The Andrews-Gordon identities and $q$-multinomial coefficients

We prove polynomial boson-fermion identities for the generating function of the number of partitions of $n$ of the form $n=\sum_{j=1}^{L-1} j f_j$, with $f_1\leq i-1$, $f_{L-1} \leq i'-1$ and $f_j+f_{j+1}\leq k$. The bosonic side of the identities involves $q$-deformations of the coefficients of $x^a$ in the expansion of $(1+x+\cdots+ x^k)^L$. A combinatorial interpretation for these $q$-multinomial coefficients is given using Durfee dissection partitions. The fermionic side of the polynomial identities arises as the partition function of a one-dimensional lattice-gas of fermionic particles. I…

Ghost级数与Bressoud-Göllnitz-Gordon恒等式的一个动机性证明

Ghost series and a motivated proof of the Bressoud-Göllnitz-Gordon identities

本文结合ghost级数与递归方法，给出Bressoud-Göllnitz-Gordon分拆恒等式的动机性证明，并期望借此理解相关扭曲顶点代数构造。

The perturbation $ϕ_{2,1}$ of the M(p,p+1) models of conformal field theory and related polynomial character identities

Using $q$-trinomial coefficients of Andrews and Baxter along with the technique of telescopic expansions, we propose and prove a complete set of polynomial identities of Rogers-Ramanujan type for M(p, p+1) models of conformal field theory perturbed by the operator $phi_{2,1}$. The bosonic form of our polynomials is closely related to corner transfer matrix sums which arise in the computation of the order parameter in the regime $1^+$ of $A_{p-1}$ dilute models. In the limit where the degree of the polynomials tends to infinity our identities provide new companion fermionic representations for …