Hall-Littlewood polynomials and characters of affine Lie algebras
Hall-Littlewood polynomials and characters of affine Lie algebras
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DOI：
10.1016/j.aim.2015.08.011
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发表时间：
2013-04
期刊：
影响因子：
--
通讯作者：
N. Bartlett;S. Warnaar
中科院分区：
文献类型：
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作者：
N. Bartlett;S. Warnaar
摘要Weyl-Kac特征标公式给出了Kac-Moody代数的可积最高权模的特征标的一个漂亮的闭式表示。然而，它不是一个本质上是组合性的公式，模糊了正性。本文用Hall-Littlewood多项式理论证明了仿射李代数Cn(1)、A2n(2)和Dn+1(2)的某些最高权模的特征的Littlewood组合公式。通过专门化，得到了麦克唐纳恒等式的Bn(1)、Cn(1)、A2n−1(2)、A2n(2)和Dn+1(2)的推广。这些推广的ETA函数恒等式包括Rogers-Ramanujan、Andrews-Gordon和Göllnitz-Gordon Q-级数作为特殊的低阶情况。
Abstract The Weyl–Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac–Moody algebras. It is not, however, a formula that is combinatorial in nature, obscuring positivity. In this paper we show that the theory of Hall–Littlewood polynomials may be employed to prove Littlewood-type combinatorial formulas for the characters of certain highest weight modules of the affine Lie algebras C n (1), A 2 n (2) and D n+ 1 (2). Through specialisation this yields generalisations for B n (1), C n (1), A 2 n− 1 (2), A 2 n (2) and D n+ 1 (2) of Macdonald's identities for powers of the Dedekind eta-function. These generalised eta-function identities include the Rogers–Ramanujan, Andrews–Gordon and Göllnitz–Gordon q-series as special, low-rank cases.