Cylindric partitions, r ?> characters and the Andrews–Gordon–Bressoud identities
Cylindric partitions, r ?> characters and the Andrews–Gordon–Bressoud identities
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圆柱分区、 r ?> 字符和 Andrews-Gordon-Bressoud 恒等式
DOI：
10.1088/1751-8113/49/16/164004
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
T A Welsh
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作者：
O. Foda;T A Welsh
我们研究了Andrews-Gordon-Bressoud（AGB）推广的Rogers-Ramanujan q-级数恒等式在圆柱分区的背景下。我们回顾的r-圆柱分区的定义，并提供了一个简单的证明鲍罗丁的产品表达其生成函数，可以被视为一个限制情况下的未发表的证明Krattenthaler。我们还回顾了r-圆柱配分生成函数之间的关系，sl？r？>代数，R，r + d？>的最小模型特征代数，和r-字符串abaci生成函数，为每个提供简单的证明。然后，我们设置r = 2，并使用两个圆柱分区重新推导AGB恒等式如下。首先，我们使用Borodin的乘积表达式的生成函数的两个圆柱分区与无限长的部分，以获得产品的AGB身份，倍因子（q ; q）∞ − 1？>，这是普通分区的生成函数。接下来，我们得到一个双射从两个圆柱分区，通过两个字符串abaci，到装饰版本的Bressoud的限制格路径。推广Bressoud的方法之间的转换，遵守不同的限制，限制路径，我们得到的总和表达式与显着非负系数的两个圆柱分区的生成函数，其中包含一个因素（q ; q）∞ − 1？> .使相同的两圆柱分拆的乘积和和表达式相等，并取消因子（q ; q）∞ − 1？>在每一侧，我们获得AGB恒等式。
We study the Andrews–Gordon–Bressoud (AGB) generalisations of the Rogers–Ramanujan q-series identities in the context of cylindric partitions. We recall the definition of r-cylindric partitions, and provide a simple proof of Borodin’s product expression for their generating functions, that can be regarded as a limiting case of an unpublished proof by Krattenthaler. We also recall the relationships between the r-cylindric partition generating functions, the principal characters of sl ˆ r ?> algebras, the r r , r + d ?> minimal model characters of r ?> algebras, and the r-string abaci generating functions, providing simple proofs for each. We then set r = 2, and use two-cylindric partitions to re-derive the AGB identities as follows. Firstly, we use Borodin’s product expression for the generating functions of the two-cylindric partitions with infinitely long parts, to obtain the product sides of the AGB identities, times a factor ( q ; q ) ∞ − 1 ?> , which is the generating function of ordinary partitions. Next, we obtain a bijection from the two-cylindric partitions, via two-string abaci, into decorated versions of Bressoud’s restricted lattice paths. Extending Bressoud’s method of transforming between restricted paths that obey different restrictions, we obtain sum expressions with manifestly non-negative coefficients for the generating functions of the two-cylindric partitions which contains a factor ( q ; q ) ∞ − 1 ?> . Equating the product and sum expressions of the same two-cylindric partitions, and canceling a factor of ( q ; q ) ∞ − 1 ?> on each side, we obtain the AGB identities.
Michio Jimbo：“椭圆量子群的准霍普夫扭曲器”变换群（即将公布）。
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