Crystal bases and q-identities

Crystal bases, dilogarithm identities and torsion in algebraic K-groups

We give a new interpretation and proof of the dilogarithm identities, associated to the affine Kac-Moody algebra sl(2)^, using the path description of the corresponding crystal basis. We also discuss connections with algebraic K-theory.

Some crystal Rogers-Ramanujan type identities

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.

通过晶体能量得到的 Rogers–Ramanujan 恒等式的 Fibonacci 变体

A Fibonacci variant of the Rogers-Ramanujan identities via crystal energy

本文为完美晶体引入长度函数，并利用晶体能量导出含 Fibonacci 数 q-模拟的 Rogers–Ramanujan 恒等式变体。

来自 $A_1^{(1)}$ 高层晶体的分拆恒等式

Partition identities from higher level crystals of $A_1^{(1)}$

本文利用多重基分拆理论研究了仿射李代数 $A_1^{(1)}$ 各层标准模的完美晶体，证明了一族带有绝对值差条件、与 Andrews-Gordon 及 Meurman-Primc 恒等式相关的分拆恒等式。

Generalisations of Capparelli's and Primc's identities, II: perfect $A_{n-1}^{(1)}$ crystals and explicit character formulas

In the first paper of this series, we gave infinite families of coloured partition identities which generalise Primc's and Capparelli's classical identities. In this second paper, we study the representation theoretic consequences of our combinatorial results. First, we show that the difference conditions we defined in our $n^2$-coloured generalisation of Primc's identity, which have a very simple expression, are actually the energy function with values in $\{0,1,2\}$ for the perfect crystal of the tensor product of the vector representation and its dual in $A_{n-1}^{(1)}$. Then we introduce a…

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…