A triple product identity for Schur functions. (English) Zbl 0763.33008
The author proves and explores the consequences of a \(q\)-binomial theorem for Schur functions:
\[
\prod_{i=1}^ n {(ax_ i)_{\infty} \over (x_ i)_{\infty}} = \sum \left( \prod_{y\in\lambda} {1-aq^{c(y)} \over 1 - q^{h(y)}} \right) q^{n(\lambda)}s_{\lambda}(x_ 1,\ldots,x_ n),
\]
where the sum is over all partitions \(\lambda\) of length \(\leq n\), \(n(\lambda) = \sum(i-1)\lambda_ i\), \(c(i,j) = j-i\) (the content), and \(h(i,j) = \lambda_ i - i + \lambda_ j - j +1\) (the hook length). Among the corollaries is a generalization of the triple product formula, an expansion of \(\prod_{i=1}^ n(-z_ iq)_{\infty}(-z_ i^{-1})_{\infty}(q)_{\infty}\) in terms of Schur functions.
Reviewer: D.M.Bressoud (University Park)
MSC:
| 33D10 | Basic theta functions (MSC1991) |
| 20C30 | Representations of finite symmetric groups |
| 11P82 | Analytic theory of partitions |
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