Let $W$ be a finite Coxeter group and $\mathcal{N}(W)$ its nilCoxeter algebra (over the reals, say), as defined at https://en.wikipedia.org/wiki/Nil-Coxeter_algebra. $\mathcal{N}(W)$ has a natural basis $\{u_w\colon w\in W\}$. We have a grading $$ \mathcal{N}(W)=\mathcal{N}(W)_0\oplus \mathcal{N}(W)_1\oplus \cdots, $$ where $\mathcal{N}(W)_i$ is spanned by all $u_w$ for which $\ell(w)=i$, where $\ell$ is the length function of $W$. My question is the following: does there exist an element $\theta\in \mathcal{N}(W)_1$ such that for all $0\leq i< N/2$ (where $N$ is the length of the longest element $w_0$), the map $\theta^{N-2i}\colon \mathcal{N}(W)_i\to \mathcal{N}(W)_{N-i}$ given by $v\mapsto \theta^{N-2i}v$ is a bijection?

This result (if true) is an analogue of the hard Lefschetz theorem for the cohomology ring of a complex projective variety. However, $\mathcal{N}(W)$ is not such a cohomology ring since multiplication does not have the right commutativity properties. If the result is true, then the weak order on $W$ satisfies the Sperner property, which I believe is still open even for the symmetric group.

If $\theta$ exists, then of course a generic $\theta$ will suffice. However, such results are difficult to prove for generic $\theta$. Perhaps we can simply take $\theta=\sum u_i$ (where the $u_i$'s are the generators of $\mathcal{N}(W)$). There is reason to believe that for the symmetric group $\mathfrak{S}_n$, a good choice of $\theta$ might be $\sum_{i=1}^{n-1} iu_i$.

**Addendum.** A more detailed and explicit version of this question is now available at https://arxiv.org/pdf/1704.00851.pdf.

**Second Addendum.** The problem has been solved in
http://front.math.ucdavis.edu/1812.00321,
http://front.math.ucdavis.edu/1811.05501, and
http://front.math.ucdavis.edu/1812.05126.

flagvariety $X$ on which $\mathcal N(W)$ acts (in the crystalographic case of course). In particular, it feels like it should be related to whether $\theta^{N-2i}:H^{2(N-i)}(X) \to H^{2i}(X)$ is an isomorphism. $\endgroup$ – Allen Knutson Jan 22 '17 at 10:30