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Asymptotics of degrees of someS n-sub regular representations

Authors:	Amitai Regev

Institution:	(1) Department of Theoretical Mathematics, The Weizmann Institute of Science, 76100 Rehovot, Israel;(2) Department of Mathematics, The Pennsylvania State University, 16802 University Park, PA, USA

Abstract:	The numbers $r\lambda = \sum {_{i \geqslant 1} d_{\lambda /\left( {2,1^{2i - 1} } \right)} ,\lambda }$ % MathType!End!2!1!, λ ⊢n appear in the enumeration of various objects, as well as coefficients inS _nrepresentations associated with products of higher commutators. We study their asymptotics asn→∞ and show that if (λ₁, λ₂, …)≈(α ₁,α ₂, …)n, if (λ′₁, λ′₂, …)≈(β ₁,β ₂, …)n and ifγ=1− Σ_k⩽1(α_k⩽1+β_k⩽1), then $\mathop {\lim }\limits_{x \to \infty } \frac{{r\lambda }}{{d\lambda }} = e^{ - \gamma } \prod\limits_{k \geqslant 1} {\frac{{1 - \alpha _k }}{{1 + \beta _k }}}$ % MathType!End!2!1!. Work partially supported by N.S.F. Grant No. DMS 94-01197.

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