Character formula for wreath products of compact groups with the infinite symmetric group

Takeshi Hirai; Etsuko Hirai

Displaying similar documents to “Character formula for wreath products of compact groups with the infinite symmetric group”

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Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products $\prod_{\binom{k = 1}{U_{d^{k}} \neq - a_{i}}}^{\infty} (1 + (a_{i}) / (U_{d^{k}}))$ (i=1,...,m) or $\prod_{\binom{k = 1}{V_{d^{k}} \neq - a_{i}}}^{\infty} (1 + (a_{i}) (V_{d^{k}})$ (i=1,...,m) to be algebraically dependent, where $a_{i}$ are non-zero integers and $U_{n}$ and $V_{n}$ are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers $a_{1}, . . ., a_{m}$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically...

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Suppose that $μ$ is an absolutely continuous probability measure on $^{R}$ n, for large $n$ . Then $μ$ has low-dimensional marginals that are approximately spherically-symmetric. More precisely, if $n \geq {(C / ε)}^{C d}$ , then there exist $d$ -dimensional marginals of $μ$ that are $ε$ -far from being sphericallysymmetric, in an appropriate sense. Here $C > 0$ is a universal constant.

On the average behavior of the Fourier coefficients of $j$ th symmetric power $L$ -function over certain sequences of positive integers

Anubhav Sharma, Ayyadurai Sankaranarayanan (2023)

Czechoslovak Mathematical Journal

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We investigate the average behavior of the $n$ th normalized Fourier coefficients of the $j$ th ( $j \geq 2$ be any fixed integer) symmetric power $L$ -function (i.e., $L (s, {sym}^{j} f)$ ), attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $S L (2, ℤ)$ over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum $S_{j}^{*} : = \sum_{a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} + a_{5}^{2} + a_{6}^{2} \leq x (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}) \in ℤ^{6}} λ_{{sym}^{j} f}^{2} (a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} + a_{5}^{2} + a_{6}^{2}),$ where $x$ is sufficiently large, and $L (s, {sym}^{j} f) : = \sum_{n = 1}^{\infty} \frac{λ_{{sym}^{j} f} (n)}{n^{s}} .$ When $j = 2$ , the error term which we obtain improves the earlier known result.