# Irreducible $G$-representations with unital algebra structure

Let us work over $$\mathbb C$$. Suppose that $$G$$ is a semisimple algebraic group and let $$H \subset G$$ be a maximal torus. Consider a dominant weight $$\omega$$, then one can associate a unique irreducible $$G$$-representation $$V(\omega)$$.

Consider the following example. Let $$G=\mathrm{SL}_3(\mathbb C)$$ and let $$W=\mathbb C^3$$ be the standard $$G$$-representation. We define $$V=\mathrm{Sym}^2W,$$ it is a $$6$$-dimensional irreducible representation for $$G$$. Moreover, we can also see $$V$$ as the space of symmetric $$3 \times 3$$ matrices. In particular, it inherits the structure of unital associative algebra using the matrix product. This is an example of Jordan algebra. On the other hand, if we define $$\tilde V=W \otimes W^\vee,$$ it contains an irreducible $$G$$-representation that corresponds to the Lie algebra $$\mathfrak g$$.

In fact, these are all examples of $$G$$-representations that have also an algebra structure. My question is: given a group $$G$$ and a dominant weight $$\omega$$, can we know when $$V(\omega)$$ has a unital algebra structure?

• Any algebra $V$ can be viewed as a map (the multiplication) $\mu: V \otimes V \to V$. If we don't require associativity OR any relation to $G$, then of course we can define an algebra structure on every irrep $V$ of $G$. Of course you do want some relation to $G$, I expect you want $\mu$ to be a $G$-map? Then the first step is to find out (using Weyl character formula or such) for which $V$ we have that $V$ appears as a summand in the decomposition of $V \otimes V$ into irreps. This is already a restriction. Sep 7, 2021 at 9:24
• The second issue of the resulting algebra being unital or not is trickier (that is why my two comments are comments rather than answers) but I am also not sure if we agree on what that word means. I thought it means there must be an element $e \in V$ such that $ex = xe = x$ for all $x \in V$. But this doesn't hold in the case of Lie algebras, which seems to be one of your examples Sep 7, 2021 at 9:26
• @Vincent: Yes, I want that $\mu$ is $G$-equivariant. I correct the misprint: I want the algebra to be unital in the sense that you wrote Sep 7, 2021 at 9:35
• For symmetric matrices the product is not an operation (the product of two symmetric matrices is not always symmetric). You need the symmetrized product $AB+BA$, which is not associative (but indeed Jordan).
– YCor
Sep 7, 2021 at 10:34
• Correct, the operation on $Sym^2$ is $A.X = AXA^T$, not $A.X = AXA^{-1}$ like in the case of $W\otimes W^\ast$. The latter does commute with the Jordan algebra structure, the former does not. Sep 7, 2021 at 13:26

The unit poses a problem here. If you require the multiplication map $$\cdot: V\otimes V\to V$$ to be $$G$$-equivariant, then this means $$gv\cdot gw = g(v\cdot w)$$. Letting $$w=1_V$$ and using that $$g:V\to V$$ is surjective, we see that $$g(1_V) = 1_V$$. In particular: $$\mathbb{C} 1_V$$ is a subrepresentation of $$V$$ isomorphic to the trivial representation. If $$V$$ is irreducible, this forces $$V$$ itself to be the trivial representation.