# Adjoint representation in second tensor power of faithful representation

Let $$G$$ be a semisimple Lie group and $$(\pi,V)$$ a faithful finite dimensional representation of $$G$$. Consider the action of $$G$$ on $$V \otimes V$$ by $$g \cdot (v_1 \otimes v_2)= gv_1 \otimes gv_2$$ Is it always true that one of the irreducible sub representations of $$V \otimes V$$ is the adjoint representation of $$G$$?

Thoughts so far:

This is true for $$G=SU_n$$ and the natural $$n$$ dimensional faithful representation. Indeed in this case $$V \otimes V$$ decomposes as a direct sum of the adjoint representation (dimension $$n^2-1$$) with one copy of the trivial representation (dimension 1).

• This is not true even for $GL$ and the natural module $V$. $V\otimes V^*$ decomposes as the sum of the adjoint and the trivial. $V\otimes V$ always has $\Lambda^2(V)$ and $S^2(V)$ as submodules, or dimensions $n(n-1)/2$ and $n(n+1)/2$. It will rarely hold for small-dimensional modules $V$, is my guess. Commented Jun 17, 2022 at 21:57
• @DavidA.Craven is it true if we require G to be semisimple? Or are there still obvious counterexamples? Commented Oct 13, 2022 at 1:39
• @IanGershonTeixeira The claim isn't true for $SL(3, \mathbb R)$ and the standard representation (call it $V$), for example. Indeed, decomposing into irreducible subrepresentations gives $V \otimes V \cong S^2 V \oplus \bigwedge^2 V \cong S^2 V \oplus V^*$, but these subreps have dimension $6, 3$, resp., whereas the adjoint representation $\mathcal{sl}_3(\Bbb R)$ has dimension $3^2 - 1 = 8$. Commented Oct 13, 2022 at 2:06
• Are you sure you didn't mean to ask about $V \otimes V^{\ast}$? This always contains $\mathfrak{g}$ as a subrepresentation, given by the action map $\mathfrak{g} \to \text{End}(V)$. Commented Oct 13, 2022 at 3:32
• @MarianoSuárez-Álvarez Ya I was just confused/didn't do a careful job reading. Other papers referred me to arxiv.org/pdf/math/0502080.pdf and the introduction there mentions small tensor powers of $V$ like $V^{\otimes 2}$ but in retrospect the actual result relevant for me is Theorem 1.5, which is about $V \otimes V^*$, as David A. Craven immediately and Qiaochu Yuan yesterday both suspected. Commented Oct 13, 2022 at 23:02

The claim is false in general; see below for some counterexamples for simple $$G$$. As mentioned in the comments, by definition any faithful representation $$V$$ of $$\mathfrak{g}$$ defines an embedding $$\mathfrak{g} \hookrightarrow \operatorname{End} V \cong V \otimes V^*$$ of the adjoint representation, $$\mathfrak{g}$$, that is, $$V \otimes V^*$$ always contains an isomorphic copy of the adjoint representation.
In the case that $$V$$ admits a nondegenerate $$G$$-invariant bilinear form $$\beta$$, that form defines an isomorphism of $$G$$-representations $$V \to V^*$$ by $$v \mapsto \beta(v, \,\cdot\,)$$, i.e.,"raising an index", and so in that case $$V \otimes V \cong V \otimes V^*$$, and in particular $$V \otimes V$$ contains a copy of $$\mathfrak{g}$$. This condition applies, for example, to all faithful representations of simple Lie groups of type $$B, C, F$$, and $$G$$, as well as $$E_7$$ and $$E_8$$.
• For $$\operatorname{SL}_n(\Bbb C)$$, $$n > 2$$, and its standard standard representation $$V$$, the tensor square decomposes into irreducibles as $$V \otimes V \cong \operatorname{Sym}^2 V \oplus \bigwedge\!{}^2 \,V ,$$ neither summand of which is the adjoint representation. (The decomposition still holds for $$n = 2$$, but in that case the adjoint representation is $$\mathfrak{sl}_2(\mathbb C) \cong \operatorname{Sym}^2 V$$.)
• For $$G = \operatorname{SO}_{4 m + 2}(\mathbb C)$$, $$m > 0$$, and $$V$$ and $$S_\pm$$ the standard and positive (negative) spin representation, respectively, the tensor square of $$S_\pm$$ decomposes into irreducibles as $$S_\pm \otimes S_\pm \cong V \oplus \bigwedge\!^3 V \oplus \cdots \oplus \bigwedge\!^{2 m - 3} V \oplus \bigwedge\!^{2 m - 1} V \oplus \bigwedge\!^{2 m + 1}_\pm V,$$ where $$\bigwedge\!^{2m + 1}_\pm V$$ is the representation of (anti-)self-dual $$(2m + 1)$$-forms, but the adjoint representation is $$\mathfrak{so}_{4 m + 2}(\mathbb C) \cong \bigwedge\!^2 V$$ .
• For a more exotic example, take $$G$$ to be the exceptional complex simple Lie group $$\operatorname{E}_6$$, and let $$V$$ denote either of the $$27$$-dimensional fundamental representations. Then, the tensor square of $$V$$ decomposes into irreducibles as: $$V \otimes V = W \oplus V^* \oplus \operatorname{Sym}^2 V ,$$ where $$W$$ is the (only) $$351$$-dimensional subrepresentation of $$\bigwedge^2 V$$. (The representation $$W$$ is itself also a counterexample, but the decomposition of $$W \otimes W$$ into irreducibles is more complicated.)