How to prove that, in a Nilcoxeter algebra $N_n$, for $ k > \frac{n(n-1)}{2}$, the product of $k$ basis elements is always zero?
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I assume that by "basis elements" you mean "standard generators" (i.e., transpositions) or at least "nontrivial elements", because otherwise, the product of any number of identity elements is nonzero. How do you define the nilCoxeter algebra? If it's using lengths of permutations, look at the lengths of $a_1a_2...a_i$ for all $i$ from $0$ to $k$ (where $a_1,a_2,...,a_k$ are your $k$ generators). If the product $a_1a_2...a_k$ is to be nonzero, these lengths have to increase by $1$ whenever $i$ grows by at least $1$, but a length cannot be greater than $\frac{n(n-1)}{2}$. – darij grinberg Sep 26 '14 at 22:15