Where do stable Kronecker coefficients live "in nature"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:41:55Z http://mathoverflow.net/feeds/question/19179 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19179/where-do-stable-kronecker-coefficients-live-in-nature Where do stable Kronecker coefficients live "in nature"? Steven Sam 2010-03-24T06:37:31Z 2010-03-28T19:16:05Z <p>Background:</p> <p>For a partition $\lambda$, let $\lambda[N] = (N - |\lambda|, \lambda_1, \lambda_2, \lambda_3, \dots)$, also let $\chi_\lambda$ be the corresponding irreducible character of the symmetric group $S_{|\lambda|}$. Even if $\lambda[N]$ is not a partition, we can make sense of $\chi_{\lambda[N]}$ as a class function of $S_N$ using a determinant, and Murnaghan proved that there exist coefficients $G^\nu_{\lambda, \mu}$ (the stable, or reduced, Kronecker coefficients) such that</p> <p>$\chi_{\lambda[N]} \chi_{\mu[N]} = \sum_\nu G^\nu_{\lambda, \mu} \chi_{\nu[N]}$</p> <p>for all $N \ge 0$. </p> <p>Question:</p> <p>In a sense, the construction of $\lambda[N]$ is a bit ad hoc. Is there a more representation-theoretic way to define these coefficients? In particular, if $|\lambda| + |\mu| = |\nu|$, then $G^\nu_{\lambda, \mu}$ coincides with the corresponding Littlewood-Richardson coefficient, so I am hopeful there is some connection. I am looking for an answer which addresses the following point: as I have defined these coefficients, it seems that the most accessible way to work with these coefficients is to use combinatorics. If I wanted to use tools from say, invariant theory or algebraic geometry, what is a more natural context for these coefficients to appear?</p>