Timeline for Finite dimensional irreducible representations of $\frak{sl}_m$ with non-trivial zero weight spaces

4 events

when toggle format	what		by	license	comment
Nov 9, 2023 at 18:32	comment	added	David E Speyer		In this language, $(\mu_1, \mu_2, \ldots, \mu_m)$ is a positive combination of the $\pi_j$ iff $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_m$, and $(\mu_1, \mu_2, \ldots, \mu_m)$ is an integer combination of the $\alpha_j$ iff $\sum \mu_j \equiv 0 \bmod m$. So representations of of $\mathfrak{sl}_m$ with a $0$-weight space correspond to partitions with at most $m$ parts and size $0 \bmod m$, modulo $(1,1,\ldots,1)$.
Nov 9, 2023 at 18:30	comment	added	David E Speyer		It's probably worth saying what this looks like in terms of the standard partition language. Weights of $\mathfrak{sl}_m$ are often described using $m$-tuples $(\mu_1, \mu_2, \ldots, \mu_m)$ modulo $\mathbb{Z}(1,1,\ldots, 1)$. (Concretely, $\mu$ corresponds to the character $\sum \mu_j t_{jj}$ of the Cartan algebra of diagonal matrices in $\mathfrak{sl}_m$. Since $\sum t_{jj}=0$, we quotient by $\mathbb{Z}(1,1,\ldots, 1)$.) The simple root $\alpha_j$ is $e_j - e_{j+1} \in \mathbb{Z}^n/\mathbb{Z}(1,1,\ldots, 1)$; the fundamental weight $\pi_j$ is $e_1+e_2+\cdots + e_j$. (continued)
Nov 9, 2023 at 18:10	vote	accept	Béla Fürdőház
Nov 9, 2023 at 17:46	history	answered	Sam Hopkins	CC BY-SA 4.0