https://numath.com/wiki/ 2026-05-05T20:55:34+0000 https://numath.com/wiki/ https://numath.com/wiki/lib/tpl/numathtemplate/images/favicon.ico text/html 2019-10-07T19:51:08+0000 Anonymous (anonymous@undisclosed.example.com) https://numath.com/wiki/doku.php?id=open-problems:asymptotic-enumeration-of-lie-group-representations&rev=1570477868&do=diff Asymtptotic enumeration of Lie group representations Statement of the problem For a simple, connected Lie group $G$, let $r_G(n)$ denote the number of $n$-dimensional representations of $G$. Find an asymptotic formula for $r_G(n)$ as $n\to\infty$. Background An asymptotic formula is known in the cases $G=SU(2)$$G=SU(3)$$SU(2)$$r_{SU(2)} = p(n)$$n$$$ p(n) \sim \frac{1}{4\sqrt{3}n} e^{\pi \sqrt{2n/3} }. $$$SU(3)$$$ r_{SU(3)} \sim \frac{K}{n^{3/5}} \exp\left(A_1 n^{2/5} - A_2 n^{3/10} - A_3 n^{… text/html 2019-10-08T22:00:31+0000 Anonymous (anonymous@undisclosed.example.com) https://numath.com/wiki/doku.php?id=open-problems:family-of-constant-term-identities&rev=1570572031&do=diff A family of constant term identities related to loop percolation Statement of the problem Define numbers $A_n$ by $$ A_n = \frac{1!4!7!\cdots (3n-2)!}{n!(n+1)!\cdots(2n-1)!}. $$ Define a multivariate polynomial $P_n(z_1,\ldots,z_n)$ by $$ P_n(z_1,\ldots,z_n) = \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j) \prod_{j=2}^n (1+z_j). $$ Zinn-Justin and Di Francesco (2008) discovered and proved (with a little help from Zeilberger (2007)) the constant term identity $$ [z_1^0 z_2^2 \cdots z_n^{2n-2}]… text/html 2019-10-07T19:55:16+0000 Anonymous (anonymous@undisclosed.example.com) https://numath.com/wiki/doku.php?id=open-problems:sample&rev=1570478116&do=diff Name of the open problem Statement of the problem The statement of the problem. E.g.: “Prove that every positive even integer greater than $2$ is a sum of two primes.” Background and history Describe where the problem comes from, what motivated it, any relevant definitions and earlier results, etc. text/html 2019-10-08T22:07:19+0000 Anonymous (anonymous@undisclosed.example.com) https://numath.com/wiki/doku.php?id=open-problems:start&rev=1570572439&do=diff Open Problems This namespace will be used for open problems. You can add your own problems here – they do not need to be well-known or named problems with a long history. * Asymptotic enumeration of Lie group representations * A family of constant term identities related to loop percolation ---------- Create a new open problem page: * After you create a page, the link to it needs to be manually added to the list above by editing this page. Automatic listing of topics will be added at…