Let $\omega=\sum\_{i=1}^n dx\_i\wedge dy\_i\in\bigwedge^2(\mathbb{R}^{2n})^\*$ be a standard symplectic form. The following result is due to Lefschetz:



> 
> For $k\leq n$, the Lefschetz operator
> $L^{n-k}:\bigwedge^k(\mathbb{R}^{2n})^\*\to
>  \bigwedge^{2n-k}(\mathbb{R}^{2n})^\*$ defined by $$
>  L^{n-k}\alpha=\alpha\wedge\underbrace{\omega\wedge\ldots\wedge\omega}\_{n-k}=\alpha\wedge\omega^{n-k}
> $$ is an isomorphism.
> 
> 
> 


This follows from Proposition 1.2.30 [1]. The proof is slick and it uses representation theory.



> 
> **Question:** What are other standard references for this result? I want to quote it properly and I would like to see a reference to a
> proof by brute force. A proof by brute force is not difficult but ugly
> and perhaps in some reference I can find an elegant presentation of
> such a proof.
> 
> 
> 


This result seems so fundamental that it should be available in many textbooks, but I am not aware of any except [1].


[1] **D. Huybrechts,** [*Complex geometry. An introduction.*](https://link.springer.com/book/10.1007/b137952). Universitext. Springer-Verlag, Berlin, 2005.
[(MathsciNet review)](https://mathscinet-ams-org.pitt.idm.oclc.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=JOUR&pg7=REFF&pg8=ET&review_format=html&s4=Huybrechts&s5=&s6=&s7=&s8=Books&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=4&mx-pid=2093043).



The Lefschetz operator
There is an elementary proof in our 2003 book *Exterior Differential Systems and Euler-Lagrange Partial Differential Equations* (Bryant, et al, University of Chicago Press). It does not use any representation theory and is not 'brute force'; it only takes a couple of paragraphs using elementary facts about exterior algebra. See Proposition 1.1a, with the proof on page 13. 


I learned that proof from Eugenio Calabi more than 30 years ago, and he told me that he had found it sometime back in the 50s.


