This is the little brother of question 68071: elementary, simple-looking and probably much easier to answer. Of course, it is just a small part of question 68071, as anybody with $\lambda$-rings experience will see.

Let $n$ and $m$ be positive integers. Let $k$ be a field of characteristic $0$. Let $U$ be the canonical $n$-dimensional representation of $\mathrm{GL}_n\left(k\right)$. Let $V$ be the canonical $m$-dimensional representation of $\mathrm{GL}_m\left(k\right)$. A folk result I cannot prove tells me that

$\wedge^2\left(U\otimes V\right)\oplus \left(\wedge^2\left(U\right)\otimes\wedge^2\left(V\right)\right)^{\oplus 2} \cong U^{\otimes 2}\otimes\wedge^2 V \oplus \wedge^2 U\otimes V^{\otimes 2}$

(don't confuse direct powers with tensor powers in this equation; also, the usual precedence rules apply where $\oplus$ is seen as addition and $\otimes$ as multiplication) as representations of $\mathrm{GL}_n\left(k\right)\times \mathrm{GL}_m\left(k\right)$.

Since the representation theory of $\mathrm{GL}_n\left(k\right)\times \mathrm{GL}_m\left(k\right)$ is semisimple (or at least I hope so?), this isomorphism must somehow "factor" into isomorphisms of parts. Or not? I am far from feeling safe here.

Anyway, is there a nice explicit description of the above isomorphism, that shows us what it maps stuff to, like the Clebsch-Gordan formulae for $\mathrm{SL}_2\left(k\right)$ ?

This is the little brother of question 68071: elementary, simple-looking and probably much easier to answer. Of course, it is just a small part of question 68071, as anybody with $\lambda$-rings experience will see.

Let $n$ and $m$ be positive integers. Let $k$ be a field of characteristic $0$. Let $U$ be the canonical $n$-dimensional representation of $\mathrm{GL}_n\left(k\right)$. Let $V$ be the canonical $m$-dimensional representation of $\mathrm{GL}_m\left(k\right)$. A folk result I cannot prove tells me that

$\wedge^2\left(U\otimes V\right)\oplus \left(\wedge^2\left(U\right)\otimes\wedge^2\left(V\right)\right)^{\oplus 2} \cong U^{\otimes 2}\otimes\wedge^2 V \oplus \wedge^2 U\otimes V^{\otimes 2}$

(don't confuse direct powers with tensor powers in this equation; also, the usual precedence rules apply where $\oplus$ is seen as addition and $\otimes$ as multiplication) as representations of $\mathrm{GL}_n\left(k\right)\times \mathrm{GL}_m\left(k\right)$.

Since the representation theory of $\mathrm{GL}_n\left(k\right)\times \mathrm{GL}_m\left(k\right)$ is semisimple (or at least I hope so?), this isomorphism must somehow "factor" into isomorphisms of parts. Or not? I am far from feeling safe here.

Anyway, is there a nice explicit description of the above isomorphism, that shows us what it maps stuff to, like the Clebsch-Gordan formulae for $\mathrm{SL}_2\left(k\right)$ ?

This is the little brother of question 68071: elementary, simple-looking and probably much easier to answer. Of course, it is just a small part of question 68071, as anybody with $\lambda$-rings experience will see.

Let $n$ and $m$ be positive integers. Let $k$ be a field of characteristic $0$. Let $U$ be the canonical $n$-dimensional representation of $\mathrm{GL}_n\left(k\right)$. Let $V$ be the canonical $m$-dimensional representation of $\mathrm{GL}_m\left(k\right)$. A folk result I cannot prove tells me that

$\wedge^2\left(U\otimes V\right)\oplus \left(\wedge^2\left(U\right)\otimes\wedge^2\left(V\right)\right)^2 \cong U^{\otimes 2}\otimes\wedge^2 V \oplus \wedge^2 U\otimes V^{\otimes 2}$

(don't confuse direct powers with tensor powers in this equation; also, the usual precedence rules apply where $\oplus$ is seen as addition and $\otimes$ as multiplication) as representations of $\mathrm{GL}_n\left(k\right)\times \mathrm{GL}_m\left(k\right)$.

Since the representation theory of $\mathrm{GL}_n\left(k\right)\times \mathrm{GL}_m\left(k\right)$ is semisimple (or at least I hope so?), this isomorphism must somehow "factor" into isomorphisms of parts. Or not? I am far from feeling safe here.

Anyway, is there a nice explicit description of the above isomorphism, that shows us what it maps stuff to, like the Clebsch-Gordan formulae for $\mathrm{SL}_2\left(k\right)$ ?

This is the little brother of question 68071: elementary, simple-looking and probably much easier to answer. Of course, it is just a small part of question 68071, as anybody with $\lambda$-rings experience will see.

Let $n$ and $m$ be positive integers. Let $k$ be a field of characteristic $0$. Let $U$ be the canonical $n$-dimensional representation of $\mathrm{GL}_n\left(k\right)$. Let $V$ be the canonical $m$-dimensional representation of $\mathrm{GL}_m\left(k\right)$. A folk result I cannot prove tells me that

$\wedge^2\left(U\otimes V\right)\oplus \left(\wedge^2\left(U\right)\otimes\wedge^2\left(V\right)\right)^{\oplus 2} \cong U^{\otimes 2}\otimes\wedge^2 V \oplus \wedge^2 U\otimes V^{\otimes 2}$

(don't confuse direct powers with tensor powers in this equation; also, the usual precedence rules apply where $\oplus$ is seen as addition and $\otimes$ as multiplication) as representations of $\mathrm{GL}_n\left(k\right)\times \mathrm{GL}_m\left(k\right)$.

Since the representation theory of $\mathrm{GL}_n\left(k\right)\times \mathrm{GL}_m\left(k\right)$ is semisimple (or at least I hope so?), this isomorphism must somehow "factor" into isomorphisms of parts. Or not? I am far from feeling safe here.

Anyway, is there a nice explicit description of the above isomorphism, that shows us what it maps stuff to, like the Clebsch-Gordan formulae for $\mathrm{SL}_2\left(k\right)$ ?