I remember some years ago reading about "lambda rings" learning there are some relations between exterior powers and their combinations. The smallest example of such a relation was $\Lambda^2\Lambda^2V\oplus\Lambda^4V\cong V\otimes\Lambda^3V$ (and there was another involving I think $\Lambda^2(V\otimes W)$ but I don't remember what it was). Checking characters I verified these must be equivalent representations of $\mathrm{GL}(V)$, so in other words $\Lambda^2\Lambda^2\oplus\Lambda^4$ must be "naturally isomorphic" to $\mathrm{Id}\otimes\Lambda^3$.
What is such a natural isomorphism, explicitly?
The first thing that came to mind was to examine the "decategorified" situation. An ordered basis $\{v_i\}$ for $V$ lends to ordered bases for these reps, and if the natural isomorphism sends such bases for the one rep into such bases for the other (an optimistic guess), then the basis ought to be equivalent as $S_n$-sets (where $n=\dim V$). To this end:
- $\Lambda^4V$ (irrep) has an ordered basis $\{v_i\wedge w_j\wedge v_k\wedge v_l\mid {\small i<j<k<l}\}$
- $\Lambda^2\Lambda^2V$ has an ordered basis consisting of two parts: $$\{(v_i\wedge v_j)\wedge(v_k\wedge v_l)\mid {\small i<j<k<l}\} \\ \sqcup \\ \{(v_i\wedge v_j)\wedge(v_i\wedge v_k)\mid {\small j<k;\, i\not\int\{j,k\}}\}$$
- $V\otimes\Lambda^3V$ has an ordered basis consisting of two parts: $$\{v_i\otimes(v_j\wedge v_k\wedge v_l)\mid {\small j<k<l;\,i\not\in\{j,k,l\}}\} \\ \sqcup \\ \{v_i\otimes(v_i\wedge v_j\wedge v_k)\mid {\small j<k;\, i\not\in\{j,k\}}\} $$
These reveal that, as $S_n$-sets, one basis has $3$ orbits while the other has $2$, so a natural isomorphism must combine these basis vectors into mixtures somehow.
I'd be fine with a natural isomorphism that takes for granted $V$ is an inner product space. (A mere vector space isomorphism is just a simple combinatorics exercise so that wouldn't cut it.)
[Also, any direct reference to more of these relations and how to generate them, with the least amount of literature-brushing-up as a prerequisite?]
[I'd also be curious if $\binom{\binom{n}{2}}{2}+\binom{n}{4}-n\binom{n}{3}$, suitably interpreted as a virtual $S_n$-set, has a special place in the Burnside ring of $S_n$, or maybe the ideal generated by all lambda relations has some special description, or whatever.]