# A hard Lefschetz theorem for nilCoxeter algebras

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.

• Do you like to have $0\leq i\leq\lfloor\frac{N}2\rfloor$? It might be complete when $N$ is odd. – T. Amdeberhan Jan 19 '17 at 17:23
• @T.Amdeberhan: when $N$ is even and $i=N/2$, the map $\theta^{N-2i}$ is the identity, so no problem. – Richard Stanley Jan 19 '17 at 17:26
• I'd say it's more closely analogous to hard Lefschetz for the cohomology of the flag variety $X$ 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. – Allen Knutson Jan 22 '17 at 10:30
• Oops, I meant "complex projective variety," not "projective toric variety." This has been corrected. – Richard Stanley Jan 23 '17 at 1:20