Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

The following may be known, but I didn't find anything in the literature.

Background:

The irreducible representations of $S_n$ correspond to shapes of Young tableaux with $n$ elements. Let $\lambda$ be a shape of a Young tablaux. The corresponding irreducible representation is generated by the set of all $e_T$, where $T$ is a Young tableaux of shape $\lambda$, and $e_T$ is the polytabloid corresponding to $T$. A basis for the irreducible representation is given by all standard Young tableaux $T_1,\ldots,T_m$ of shape $\lambda$. They in fact span it as a lattice. That it, for any tableaux $T$ of shape $\lambda$, $$ e_T = c_1 e_{T_1} + \ldots + c_m e_{T_m} $$ where $c_1,\ldots,c_m \in \mathbb{Z}$.

Question:

How large are the coefficients $c_1,\ldots,c_m$ (in absolute value)?

A simple analysis of the straightening algorithm (see for example en.wikipedia.org/wiki/Garnir_relations, or the book of Sagan) gives a bound of $n^{O(n^2)}$, but I would be surprised if this is tight. Can it be that the coefficients are bounded by $n^{O(n)}$ or even $\exp(n)$?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.